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"Times" 1 12 0 0 0 1 2 2 1 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 27 "More differentiation rules " }} {PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }} {PARA 0 "" 0 "" {TEXT -1 18 "Version: 26.9.2006" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 48 "The chain rule for differentiation - explanation" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 64 "Consider the problem of finding the derivative of the function \+ " }{XPPEDIT 18 0 "f(x)=sqrt(1-x^2)" "6#/-%\"fG6#%\"xG-%%sqrtG6#,&\"\" \"F,*$F'\"\"#!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 69 "We \+ can split up the evaluation of f into two stages: first calculate " } {XPPEDIT 18 0 "1-x^2" "6#,&\"\"\"F$*$%\"xG\"\"#!\"\"" }{TEXT -1 32 ", \+ and then take the square root." }}{PARA 0 "" 0 "" {TEXT -1 58 "Introdu ce the variable u to denote the intermediate value " }{XPPEDIT 18 0 "1 -x^2" "6#,&\"\"\"F$*$%\"xG\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 11 "If we let " }{XPPEDIT 18 0 "y = f(x)" "6#/%\"yG-%\"fG 6#%\"xG" }{TEXT -1 15 ", we now have " }{XPPEDIT 18 0 "u=1-x^2" "6#/% \"uG,&\"\"\"F&*$%\"xG\"\"#!\"\"" }{TEXT -1 6 " and " }{XPPEDIT 18 0 " y=sqrt(u)" "6#/%\"yG-%%sqrtG6#%\"uG" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 488 205 205 {PLOTDATA 2 "65-%'CURVESG6$7$ 7$$!\"*\"\"!$F*F*7$$!\"'F*F+-%'COLOURG6&%$RGBGF*F*F*-F$6$7$7$$!\"#F*F+ 7$$\"\"#F*F+F/-F$6$7$7$$\"\"'F*F+7$$\"\"*F*F+F/-F$6$7%7$$!3G********** ***>)!#<$\"35+++++++?!#=7$$!3#)*************z(FKF+7$FI$!35+++++++?FNF/ -F$6$7%7$$\"3#)*************z(FKFL7$$\"3G*************>)FKF+7$FYFSF/-F $6$7%7$FSFL7$FLF+7$FSFSF/-%)POLYGONSG6%7&7$F:F:7$F@F:7$F@F77$F:F77&7$F 7F:7$F7F77$F-F77$F-F:-%&COLORG6&F2$\"\"(!\"\"F_p$FDFap-F$6$7'7$$!\"(F* $\"\"$F*7$$F`pF*Fip7$F\\q$!\"$F*7$FgpF^qFfp-F]p6&F2\"\"\"F*F*-%%TEXTG6 %7$$\"\"%F*$FjpFapQ)take~the6\"-%%FONTG6$%*HELVETICAGFD-Feq6%7$Fhq$F_q FapQ,square~rootF\\rF]r-Feq6%7$$!\"%F*FjqQ+square~andF\\rF]r-Feq6%7$Fi rFdrQ0subtract~from~1F\\rF]r-Feq6%7$$!#&*FapF+Q\"xF\\rF]r-Feq6%7$F+F_p Q\"uF\\rF]r-Feq6%7$$\"#&*FapF+Q\"yF\\rF]r-Feq6&7$F+$!#MFapQ\"fF\\rFaqF ]r-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6%Q!F\\rF]u-F^r6#%(DEFAULTG-%%VI EWG6$F`uF`u" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 " Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" " Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve \+ 14" "Curve 15" "Curve 16" }}{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 47 "f is the composition of the function u where " }{XPPEDIT 18 0 "u(x)=1-x^2" "6#/-%\"uG6#%\"xG,&\"\"\"F)*$F'\"\"#!\"\"" }{TEXT -1 35 " followed by the function g where " }{XPPEDIT 18 0 "g(x)=sqrt(x" "6#/-%\"gG6#%\"xG-%%sqrtG6#F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 97 "Here we are abusing notat ion by using the same letter u to denote both a function and its value . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "Sin ce we have " }{XPPEDIT 18 0 "u=1-x^2" "6#/%\"uG,&\"\"\"F&*$%\"xG\"\"# !\"\"" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "y=sqrt(u)" "6#/%\"yG-%%sqr tG6#%\"uG" }{TEXT -1 13 ", we can find" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "du/dx=-2*x" "6#/*&%#duG\"\"\"%#dxG!\"\",$*&\"\" #F&%\"xGF&F(" }{TEXT -1 9 " and " }{XPPEDIT 18 0 "dy/du=1/(2*sqrt( u))" "6#/*&%#dyG\"\"\"%#duG!\"\"*&F&F&*&\"\"#F&-%%sqrtG6#%\"uGF&F(" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 42 "We would like to be able to conclude that:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx=dy/du" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&F%F&%#duGF(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "du/dx" "6#*&%#duG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 7 "giving " }}{PARA 256 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "dy/dx = 1/(2*sqrt(u));" "6#/*&%#dyG\"\"\"%#dx G!\"\"*&F&F&*&\"\"#F&-%%sqrtG6#%\"uGF&F(" }{XPPEDIT 18 0 "``(-2*x) = - x/sqrt(1-x^2);" "6#/-%!G6#,$*&\"\"#\"\"\"%\"xGF*!\"\",$*&F+F*-%%sqrtG6 #,&F*F**$F+F)F,F,F," }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 270 17 "_________________" }{TEXT -1 1 " " }{TEXT 0 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "In order \+ to try and justify this approach, consider a small change " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT 384 1 "x" }{TEXT -1 17 " in the vari able " }{TEXT 383 1 "x" }{TEXT -1 10 ", and let " }{XPPEDIT 18 0 "Delt a;" "6#%&DeltaG" }{TEXT 385 1 "u" }{TEXT -1 5 " and " }{XPPEDIT 18 0 " Delta;" "6#%&DeltaG" }{TEXT 386 1 "y" }{TEXT -1 47 " be the correspond ing changes in the variables " }{TEXT 387 1 "u" }{TEXT -1 5 " and " } {TEXT 388 1 "y" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 12 "Then we have" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Delta*y/(De lta*x) = Delta*y/(Delta*u);" "6#/*(%&DeltaG\"\"\"%\"yGF&*&F%F&%\"xGF&! \"\"*(F%F&F'F&*&F%F&%\"uGF&F*" }{TEXT -1 2 " " }{XPPEDIT 18 0 "Delta* u/(Delta*x);" "6#*(%&DeltaG\"\"\"%\"uGF%*&F$F%%\"xGF%!\"\"" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 394 8 "________" } {TEXT -1 1 " " }{TEXT 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "For example, let " }{TEXT 389 1 "x" }{TEXT -1 13 " = 0.6 and " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{XPPEDIT 18 0 "x = 0" "6#/%\"xG\"\"!" }{TEXT -1 14 ".01, so that " }{XPPEDIT 18 0 "x+Delta;" "6#,&%\"xG\"\"\"%&DeltaGF%" }{XPPEDIT 18 0 "x = 0" "6#/% \"xG\"\"!" }{TEXT -1 4 ".61." }}{PARA 0 "" 0 "" {TEXT -1 6 "Then " } {XPPEDIT 18 0 "u=1-0" "6#/%\"uG,&\"\"\"F&\"\"!!\"\"" }{TEXT -1 1 "." } {XPPEDIT 18 0 "6^2;" "6#*$\"\"'\"\"#" }{TEXT -1 14 " = 0.64 and " } {XPPEDIT 18 0 "u+Delta;" "6#,&%\"uG\"\"\"%&DeltaGF%" }{XPPEDIT 18 0 " \+ u= 1-0" "6#/%\"uG,&\"\"\"F&\"\"!!\"\"" }{TEXT -1 1 "." }{XPPEDIT 18 0 "61^2;" "6#*$\"#h\"\"#" }{TEXT -1 10 "= 0.6279. " }}{PARA 0 "" 0 "" {TEXT -1 6 "Thus " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{XPPEDIT 18 0 "u = 0" "6#/%\"uG\"\"!" }{TEXT -1 1 "." }{XPPEDIT 18 0 "6279-0" "6#, &\"%zi\"\"\"\"\"!!\"\"" }{TEXT -1 2 ".6" }{XPPEDIT 18 0 "4 = -0" "6#/ \"\"%,$\"\"!!\"\"" }{TEXT -1 6 ".0121." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "Now " }{XPPEDIT 18 0 "y = sqrt(.64); " "6#/%\"yG-%%sqrtG6#-%&FloatG6$\"#k!\"#" }{TEXT -1 12 " = 0.8 and " }{XPPEDIT 18 0 "y+Delta;" "6#,&%\"yG\"\"\"%&DeltaGF%" }{TEXT 390 1 "y " }{TEXT -1 3 " = " }{XPPEDIT 18 0 "sqrt(.6279);" "6#-%%sqrtG6#-%&Floa tG6$\"%zi!\"%" }{TEXT -1 1 " " }{TEXT 381 1 "~" }{TEXT -1 15 " 0.7924 014134." }}{PARA 0 "" 0 "" {TEXT -1 6 "Thus " }{XPPEDIT 18 0 "Delta; " "6#%&DeltaG" }{TEXT 382 1 "y" }{TEXT 393 1 " " }{TEXT 392 1 "~" } {TEXT 391 0 "" }{TEXT -1 3 " 0." }{XPPEDIT 18 0 "7924014134-0" "6#,&\" +MT,Cz\"\"\"\"\"!!\"\"" }{TEXT -1 1 "." }{XPPEDIT 18 0 "8 = -0" "6#/ \"\"),$\"\"!!\"\"" }{TEXT -1 12 ".0075985866." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "In this numerical example , the equation" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "De lta*y/(Delta*x) = Delta*y/(Delta*u);" "6#/*(%&DeltaG\"\"\"%\"yGF&*&F%F &%\"xGF&!\"\"*(F%F&F'F&*&F%F&%\"uGF&F*" }{TEXT -1 2 " " }{XPPEDIT 18 0 "Delta*u/(Delta*x);" "6#*(%&DeltaG\"\"\"%\"uGF%*&F$F%%\"xGF%!\"\"" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 32 "appears in approximate f orm as: " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "``(`-0. 0075985866`/`0.01`) = ``(`-0.0075985866`/`-0.0121`);" "6#/-%!G6#*&%.-0 .0075985866G\"\"\"%%0.01G!\"\"-F%6#*&F(F)%(-0.0121GF+" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "``(`-0.00121`/`0.01`);" "6#-%!G6#*&%)-0.00121G\"\"\" %%0.01G!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "In particular, " }{XPPEDIT 18 0 "Delta*y/(Delt a*x);" "6#*(%&DeltaG\"\"\"%\"yGF%*&F$F%%\"xGF%!\"\"" }{TEXT -1 1 " " } {TEXT 261 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "-0" "6#,$\"\"!!\"\"" } {TEXT -1 12 ".75985866, " }{XPPEDIT 18 0 "Delta*y/(Delta*u);" "6#*(%& DeltaG\"\"\"%\"yGF%*&F$F%%\"uGF%!\"\"" }{TEXT -1 1 " " }{TEXT 262 1 "~ " }{TEXT -1 18 " 0.62798236 and " }{XPPEDIT 18 0 "Delta*u/(Delta*x); " "6#*(%&DeltaG\"\"\"%\"uGF%*&F$F%%\"xGF%!\"\"" }{TEXT -1 1 " " } {TEXT 263 1 "~" }{TEXT -1 1 " " }{XPPEDIT 18 0 "-1" "6#,$\"\"\"!\"\"" }{TEXT -1 4 ".21." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "We now have to imagine " }{XPPEDIT 18 0 "Delta;" "6#%&Del taG" }{TEXT 317 1 "x" }{TEXT -1 36 " getting smaller and smaller, with " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT 315 1 "u" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT 316 1 "y" }{TEXT -1 27 " likewise approaching zero." }}{PARA 0 "" 0 "" {TEXT -1 12 "The formula " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Delta*y/ (Delta*x) = Delta*y/(Delta*u);" "6#/*(%&DeltaG\"\"\"%\"yGF&*&F%F&%\"xG F&!\"\"*(F%F&F'F&*&F%F&%\"uGF&F*" }{TEXT -1 2 " " }{XPPEDIT 18 0 "Del ta*u/(Delta*x);" "6#*(%&DeltaG\"\"\"%\"uGF%*&F$F%%\"xGF%!\"\"" }{TEXT -1 15 ". ------- (i) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 279 14 "______________" }{TEXT -1 1 " " }{TEXT 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 51 "holds at every stage, no \+ matter how small we take " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" } {TEXT 318 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 61 "From th e definition of derivative, we would expect that, as " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT 319 1 "x" }{TEXT -1 14 " tends to 0, \+ " }{XPPEDIT 18 0 "Delta*y/(Delta*u);" "6#*(%&DeltaG\"\"\"%\"yGF%*&F$F% %\"uGF%!\"\"" }{TEXT -1 12 " tends to " }{XPPEDIT 18 0 "dy/du" "6#*& %#dyG\"\"\"%#duG!\"\"" }{TEXT -1 7 ", and " }{XPPEDIT 18 0 "Delta*u/( Delta*x);" "6#*(%&DeltaG\"\"\"%\"uGF%*&F$F%%\"xGF%!\"\"" }{TEXT -1 11 " tends to " }{XPPEDIT 18 0 "du/dx" "6#*&%#duG\"\"\"%#dxG!\"\"" } {TEXT -1 35 ", so the right hand side tends to " }{XPPEDIT 18 0 "dy/d u" "6#*&%#dyG\"\"\"%#duG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "du/dx" "6#*&%#duG\"\"\"%#dxG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 88 "On the other hand, since the formula (i) holds at every stage i n the limiting process, " }{XPPEDIT 18 0 "Delta*y/(Delta*x);" "6#*(%& DeltaG\"\"\"%\"yGF%*&F$F%%\"xGF%!\"\"" }{TEXT -1 20 " must also tend t o " }{XPPEDIT 18 0 "dy/du" "6#*&%#dyG\"\"\"%#duG!\"\"" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "du/dx" "6#*&%#duG\"\"\"%#dxG!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "This means that " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx=dy/du" "6#/*&%#dyG\"\"\"%#dxG!\" \"*&F%F&%#duGF(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "du/dx" "6#*&%#duG\"\" \"%#dxG!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 11 "also hold s." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 123 "To extend this argument to the more general setting of a function f whic h is the composition of a differentiable function " }{XPPEDIT 18 0 "u = u(x)" "6#/%\"uG-F$6#%\"xG" }{TEXT -1 46 " followed by another diffe rentiable function " }{XPPEDIT 18 0 "y = g(u)" "6#/%\"yG-%\"gG6#%\"uG " }{TEXT -1 26 ", we need to ensure that " }{XPPEDIT 18 0 "Delta;" "6 #%&DeltaG" }{TEXT 320 1 "u" }{TEXT -1 20 " is not zero, when " } {XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT 321 1 "x" }{TEXT -1 32 " is sufficiently close to zero. " }}{PARA 0 "" 0 "" {TEXT -1 56 "Provided that this can be achieved, we have the general " }{TEXT 259 10 "chain rule" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx=dy/du" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&F%F&%#d uGF(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "du/dx" "6#*&%#duG\"\"\"%#dxG!\" \"" }{TEXT -1 2 " ." }}{PARA 256 "" 0 "" {TEXT 280 7 "_______" }{TEXT -1 1 " " }{TEXT 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "Returning to the specific example, where " }{XPPEDIT 18 0 "u = 1 - x^2" "6#/%\"uG,&\"\"\"F&*$%\"xG\"\"#!\"\"" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "y = sqrt(u)" "6#/%\"yG-%%sqrtG6#%\"uG" }{TEXT -1 11 ", we have " }{XPPEDIT 18 0 "du/dx = -2*x" "6#/*&%#duG\"\"\"%#d xG!\"\",$*&\"\"#F&%\"xGF&F(" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "dy/ du = 1/(2*sqrt(u))" "6#/*&%#dyG\"\"\"%#duG!\"\"*&F&F&*&\"\"#F&-%%sqrtG 6#%\"uGF&F(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 6 "When " } {TEXT 332 1 "x" }{TEXT -1 9 " = 0.6, " }{XPPEDIT 18 0 "du/dx" "6#*&%# duG\"\"\"%#dxG!\"\"" }{TEXT -1 20 " = -1.2 and, since " }{TEXT 333 1 "u" }{TEXT -1 10 " = 0.64, " }{XPPEDIT 18 0 "dy/du = ``(1/2)*``(1/`0. 8`);" "6#/*&%#dyG\"\"\"%#duG!\"\"*&-%!G6#*&F&F&\"\"#F(F&-F+6#*&F&F&%$0 .8GF(F&" }{TEXT -1 9 " = 0.625." }}{PARA 0 "" 0 "" {TEXT -1 12 "This g ives " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 10 " = (-1.2) " }{TEXT 314 1 "." }{TEXT -1 17 " (0.625) = -0.75." }} {PARA 0 "" 0 "" {TEXT -1 78 "These values may be compared with the app roximations calculated above, namely:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "Delta*u/(Delta* x);" "6#*(%&DeltaG\"\"\"%\"uGF%*&F$F%%\"xGF%!\"\"" }{TEXT -1 1 " " } {TEXT 266 1 "~" }{TEXT -1 9 " -1.21, " }{XPPEDIT 18 0 "Delta*y/(Delta *u);" "6#*(%&DeltaG\"\"\"%\"yGF%*&F$F%%\"uGF%!\"\"" }{TEXT -1 1 " " } {TEXT 264 1 "~" }{TEXT -1 19 " 0.62798236 and " }{XPPEDIT 18 0 "Del ta*y/(Delta*u);" "6#*(%&DeltaG\"\"\"%\"yGF%*&F$F%%\"uGF%!\"\"" }{TEXT -1 1 " " }{TEXT 265 1 "~" }{TEXT -1 13 " -0.75985866." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 19 "Chain rule ex amples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 76 "Suppose that a function f is the composition of a dif ferentiable function " }{XPPEDIT 18 0 "u = u(x)" "6#/%\"uG-F$6#%\"xG " }{TEXT -1 46 " followed by another differentiable function " } {XPPEDIT 18 0 "y = g(u)" "6#/%\"yG-%\"gG6#%\"uG" }{TEXT -1 1 "." }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 373 119 119 {PLOTDATA 2 "6 3-%'CURVESG6$7$7$$!\"*\"\"!$F*F*7$$!\"'F*F+-%'COLOURG6&%$RGBGF*F*F*-F$ 6$7$7$$!\"#F*F+7$$\"\"#F*F+F/-F$6$7$7$$\"\"'F*F+7$$\"\"*F*F+F/-F$6$7%7 $$!3G*************>)!#<$\"35+++++++?!#=7$$!3#)*************z(FKF+7$FI$ !35+++++++?FNF/-F$6$7%7$$\"3#)*************z(FKFL7$$\"3G*************> )FKF+7$FYFSF/-F$6$7%7$FSFL7$FLF+7$FSFSF/-%)POLYGONSG6%7&7$F:F:7$F@F:7$ F@F77$F:F77&7$F7F:7$F7F77$F-F77$F-F:-%&COLORG6&F2$\"\"(!\"\"F_p$FDFap- F$6$7'7$$!\"(F*$\"\"$F*7$$F`pF*Fip7$F\\q$!\"$F*7$FgpF^qFfp-F06&F2$\"*+ +++\"!\")F+F+-%%TEXTG6%7$$\"\"%F*F+Q\"g6\"-%%FONTG6$%*HELVETICAGFD-Fgq 6%7$$!#5F*F+Q\"xF]rF^r-Fgq6%7$F+$\"\"\"F*Q)u~=~u(x)F]rF^r-Fgq6%7$$!\"% F*F+Q\"uF]rF^r-Fgq6%7$$\"$D\"Fap$F_qFapQ3y~=~g(u(x))~=~f(x)F]rF^r-Fgq6 &7$F+$!#MFapQ\"fF]rFaqF^r-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6%Q!F]rFh t-F_r6#%(DEFAULTG-%%VIEWG6$F[uF[u" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur ve 12" "Curve 13" "Curve 14" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 53 "Then, under suitable conditions, we have the general " }{TEXT 259 10 "chain rule" }{TEXT -1 1 ":" }{TEXT 258 2 " " }{TEXT -1 0 "" } }{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx=dy/du" "6#/*& %#dyG\"\"\"%#dxG!\"\"*&F%F&%#duGF(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "du /dx" "6#*&%#duG\"\"\"%#dxG!\"\"" }{TEXT -1 2 " ." }}{PARA 256 "" 0 "" {TEXT 282 7 "_______" }{TEXT -1 1 " " }{TEXT 0 0 "" }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 23 "In function notation, " }{XPPEDIT 18 0 " f(x) = g(u(x))" "6#/-%\"fG6#%\"xG-%\"gG6#-%\"uG6#F'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "`f '`(x) = `g '`(u(x))*`.`*`u '`(x)" "6#/-%$f~'G6#% \"xG*(-%$g~'G6#-%\"uG6#F'\"\"\"%\".GF/-%$u~'G6#F'F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 281 91 "To dif ferentiate the composition of two functions, form the product of 2 fac tors as follows" }{TEXT -1 3 ":\n\n" }{TEXT 312 10 "1st factor" } {TEXT -1 5 ": " }{TEXT 259 62 "derivative of outer function evaluat ed at inner function value" }{TEXT -1 2 "\n\n" }{TEXT 313 10 "2nd fact or" }{TEXT -1 4 ": " }{TEXT 259 28 "derivative of inner function" } {TEXT 256 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "A special case of the chain rule which occurs frequently \+ is the case where " }{XPPEDIT 18 0 "g(x)=x^r" "6#/-%\"gG6#%\"xG)F'%\"r G" }{TEXT -1 7 ", with " }{TEXT 380 1 "r" }{TEXT -1 27 " is a real num ber constant." }}{PARA 0 "" 0 "" {TEXT -1 13 "In this case " } {XPPEDIT 18 0 "f(x)=u(x)^r" "6#/-%\"fG6#%\"xG)-%\"uG6#F'%\"rG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "`f '`(x)=r*u(x)^(r-1)*`u '`(x)" "6#/-%$f~ 'G6#%\"xG*(%\"rG\"\"\")-%\"uG6#F',&F)F*F*!\"\"F*-%$u~'G6#F'F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 0 "" {TEXT 258 8 "Question" }{TEXT 283 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 7 "Given " }{XPPEDIT 18 0 "y = (x^2-5)^3;" "6#/%\"yG*$,&*$%\"xG\"\"#\" \"\"\"\"&!\"\"\"\"$" }{TEXT -1 7 ", find " }{XPPEDIT 18 0 "dy/dx" "6#* &%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 257 "" 0 "" {TEXT 269 8 "Solution" }{TEXT 284 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "u = x^2-5" "6#/%\"uG ,&*$%\"xG\"\"#\"\"\"\"\"&!\"\"" }{TEXT -1 11 ", so that " }{XPPEDIT 18 0 "y = u^3" "6#/%\"yG*$%\"uG\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 8 "We have " }{XPPEDIT 18 0 "du/dx = 2*x" "6#/*&%#duG\"\"\" %#dxG!\"\"*&\"\"#F&%\"xGF&" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "dy/du \+ = 3*u^2" "6#/*&%#dyG\"\"\"%#duG!\"\"*&\"\"$F&*$%\"uG\"\"#F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 24 "Then, by the chain rule," }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = dy/du" "6#/*& %#dyG\"\"\"%#dxG!\"\"*&F%F&%#duGF(" }{TEXT -1 2 " " }{XPPEDIT 18 0 "d u/dx = 3*u^2*`.`*2*x;" "6#/*&%#duG\"\"\"%#dxG!\"\"*,\"\"$F&*$%\"uG\"\" #F&%\".GF&F-F&%\"xGF&" }{XPPEDIT 18 0 "`` = 3*(x^2-5)^2*2*x;" "6#/%!G* *\"\"$\"\"\"*$,&*$%\"xG\"\"#F'\"\"&!\"\"F,F'F,F'F+F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 6*x*(x^2-5)^2; " "6#/%!G*(\"\"'\"\"\"%\"xGF',&*$F(\"\"#F'\"\"&!\"\"F+" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT 285 13 "Alternatively" }{TEXT -1 2 ", " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 3*(x^2-5)^2; " "6#/*&%#dyG\"\"\"%#dxG!\"\"*&\"\"$F&*$,&*$%\"xG\"\"#F&\"\"&F(F/F&" } {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" } {TEXT -1 2 " [" }{XPPEDIT 18 0 "x^2-5" "6#,&*$%\"xG\"\"#\"\"\"\"\"&!\" \"" }{TEXT -1 2 "] " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 3*(x^2-5)^2*`.`*2*x;" "6#/%!G*,\"\"$\"\"\"*$,&*$%\"xG\"\"#F'\" \"&!\"\"F,F'%\".GF'F,F'F+F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 6*x*(x^2-5)^2;" "6#/%!G*(\"\"'\"\" \"%\"xGF',&*$F(\"\"#F'\"\"&!\"\"F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Diff((x^2-5) ^3,x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$*$),&*$ )%\"xG\"\"#\"\"\"F-!\"&F-\"\"$F-F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,$*&),&*$)%\"xG\"\"#\"\"\"F+!\"&F+F*F+F)F+\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The expression " }{XPPEDIT 18 0 "(x^2-5)^3" "6#*$,&*$%\"xG\"\"# \"\"\"\"\"&!\"\"\"\"$" }{TEXT -1 14 " has the form " }{XPPEDIT 18 0 "u (x)^3" "6#*$-%\"uG6#%\"xG\"\"$" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "u(x)=x^2-5" "6#/-%\"uG6#%\"xG,&*$F'\"\"#\"\"\"\"\"&!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "u := 'u':\nDiff(u(x)^3,x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$*$)-%\"uG6#%\"xG\"\"$\"\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&)-%\"uG6#%\"xG\"\"#\"\"\"-%%diffG6$F&F)F +\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "The result is a prod uct of " }{XPPEDIT 18 0 "3*u(x)^2;" "6#*&\"\"$\"\"\"*$-%\"uG6#%\"xG\" \"#F%" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "d/dx;" "6#*&%\"dG\"\"\"%#dx G!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[u(x)]" "6#7#-%\"uG6#%\"xG" } {TEXT -1 45 ", where the factors are obtained as follows.\n" }{TEXT 286 10 "1st factor" }{TEXT -1 17 ": Substitute " }{XPPEDIT 18 0 "u = u(x);" "6#/%\"uG-F$6#%\"xG" }{TEXT -1 19 " and differentiate " } {XPPEDIT 18 0 "u^3;" "6#*$%\"uG\"\"$" }{TEXT -1 17 " with respect to \+ " }{TEXT 334 1 "u" }{TEXT -1 19 " by the power rule." }}{PARA 0 "" 0 " " {TEXT 287 10 "2nd factor" }{TEXT -1 19 ": Differentiate " } {XPPEDIT 18 0 "u = u(x);" "6#/%\"uG-F$6#%\"xG" }{TEXT -1 17 " with res pect to " }{TEXT 335 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "We can obtain the last result in a series of steps as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "u := 'u':\nDiff(u^3,u);\nval ue(%);\nsubs(u=u(x),%)*diff(u(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%%DiffG6$*$)%\"uG\"\"$\"\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, $*&\"\"$\"\"\")%\"uG\"\"#F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*( \"\"$\"\"\")-%\"uG6#%\"xG\"\"#F&-%%diffG6$F(F+F&F&" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 "Now replace u(" }{TEXT 336 1 "x" }{TEXT -1 5 ") by " }{XPPEDIT 18 0 "x^2-5" "6#,&*$%\"xG\"\"# \"\"\"\"\"&!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "u := 'u':\nDiff(u^3,u);\nval ue(%);\nsubs(u=x^2-5,%)*diff(x^2-5,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$*$)%\"uG\"\"$\"\"\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$)%\"uG\"\"#\"\"\"\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&) ,&*$)%\"xG\"\"#\"\"\"F+!\"&F+F*F+F)F+\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "Note that we obtain the same re sult, if we expand the expression before differentiating." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(x^2-5)^3=(x^2-5)^2*(x^2-5) " "6#/*$,&*$%\"xG\"\"#\"\"\"\"\"&!\"\"\"\"$*&,&*$F'F(F)F*F+F(,&*$F'F(F )F*F+F)" }{XPPEDIT 18 0 "``=(x^4-10*x^2+25)*(x^2-5)" "6#/%!G*&,(*$%\"x G\"\"%\"\"\"*&\"#5F**$F(\"\"#F*!\"\"\"#DF*F*,&*$F(F.F*\"\"&F/F*" } {TEXT -1 14 " " }}{PARA 256 "" 0 "" {TEXT -1 5 " " } {XPPEDIT 18 0 "``=x^6-10*x^4+25*x^2-5*x^4+50*x^2-125" "6#/%!G,.*$%\"xG \"\"'\"\"\"*&\"#5F)*$F'\"\"%F)!\"\"*&\"#DF)*$F'\"\"#F)F)*&\"\"&F)*$F'F -F)F.*&\"#]F)*$F'F2F)F)\"$D\"F." }{XPPEDIT 18 0 "``= x^6-15*x^4+75*x^2 -125" "6#/%!G,**$%\"xG\"\"'\"\"\"*&\"#:F)*$F'\"\"%F)!\"\"*&\"#vF)*$F' \"\"#F)F)\"$D\"F." }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\" \"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[x^6-15*x^4+75*x^2-125]= 6*x^5-60*x^3+150*x" "6#/7#,**$%\"xG\"\"'\"\"\"*&\"#:F)*$F'\"\"%F)!\"\" *&\"#vF)*$F'\"\"#F)F)\"$D\"F.,(*&F(F)*$F'\"\"&F)F)*&\"#gF)*$F'\"\"$F)F .*&\"$]\"F)F'F)F)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=6*x*(x^4-10*x^2+25)" "6#/%!G*(\"\"'\"\"\"%\"xGF',(* $F(\"\"%F'*&\"#5F'*$F(\"\"#F'!\"\"\"#DF'F'" }{XPPEDIT 18 0 "``=6*x*(x^ 2-5)^2" "6#/%!G*(\"\"'\"\"\"%\"xGF',&*$F(\"\"#F'\"\"&!\"\"F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "(x^2-5)^3;\nexpand(%);\nDiff(%,x);\nvalue(%);\nfactor (%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$),&*$)%\"xG\"\"#\"\"\"F*!\"& F*\"\"$F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$)%\"xG\"\"'\"\"\"F(*$ )F&\"\"%F(!#:*$)F&\"\"#F(\"#v!$D\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%%DiffG6$,**$)%\"xG\"\"'\"\"\"F+*$)F)\"\"%F+!#:*$)F)\"\"#F+\"#v!$D \"F+F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"&\"\"\"\"\"'*$ )F&\"\"$F(!#gF&\"$]\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&),&*$)%\" xG\"\"#\"\"\"F+!\"&F+F*F+F)F+\"\"'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{PARA 257 "" 0 "" {TEXT 258 8 "Question" }{TEXT 288 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 7 "Given " }{XPPEDIT 18 0 "y = (3*x+2)^5;" " 6#/%\"yG*$,&*&\"\"$\"\"\"%\"xGF)F)\"\"#F)\"\"&" }{TEXT -1 7 ", find " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 269 8 "Solutio n" }{TEXT 289 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "u = 3*x+2;" "6#/%\"uG,&*&\"\"$\"\"\"%\"xGF(F(\"\"#F(" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "y = u^5;" "6#/%\"yG*$%\"uG\"\"&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 8 "We have " }{XPPEDIT 18 0 "du/dx = 3;" "6#/*&%#duG\"\"\"%#dxG!\"\"\"\"$" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "dy/du = 5*u^4;" "6#/*&%#dyG\"\"\"%#duG!\"\"*&\"\"&F&*$% \"uG\"\"%F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 24 "Then, by t he chain rule," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/ dx = dy/du" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&F%F&%#duGF(" }{TEXT -1 2 " \+ " }{XPPEDIT 18 0 "du/dx = 5*u^4*`.`*3;" "6#/*&%#duG\"\"\"%#dxG!\"\"** \"\"&F&*$%\"uG\"\"%F&%\".GF&\"\"$F&" }{XPPEDIT 18 0 "`` = 5*(3*x+2)^4* `.`*3;" "6#/%!G**\"\"&\"\"\"*$,&*&\"\"$F'%\"xGF'F'\"\"#F'\"\"%F'%\".GF 'F+F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 15*(3*x+2)^4;" "6#/%!G*&\"#:\"\"\"*$,&*&\"\"$F'%\"xGF'F'\" \"#F'\"\"%F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 290 13 "Alternat ively" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 5*(3*x+2)^4;" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&\"\"&F&*$,&*& \"\"$F&%\"xGF&F&\"\"#F&\"\"%F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[3*x+2];" "6#7#,&*&\"\"$\"\"\"%\"xGF'F'\"\"#F'" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 "= " }{XPPEDIT 18 0 "5*(3*x+2)^4*`.`*3;" "6#**\"\"&\" \"\"*$,&*&\"\"$F%%\"xGF%F%\"\"#F%\"\"%F%%\".GF%F)F%" }{TEXT -1 1 " " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 15*(3*x+2)^4;" "6#/%!G*&\"#:\"\"\"*$,&*&\"\"$F'%\"xGF'F'\"\"#F'\"\"%F'" }{TEXT -1 1 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Diff((3*x+2)^5,x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$*$),&%\"xG\"\"$\"\"#\"\"\"\"\"&F,F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*$),&%\"xG\"\"$\"\"#\"\"\"\"\"%F*\"#:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "Note that we obtain the same result, if we expand the expression before differe ntiating." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "(3*x+2)^5;\nexpand(%);\nDiff(%,x);\nvalue(%);\nfactor (%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$),&%\"xG\"\"$\"\"#\"\"\"\"\" &F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*$)%\"xG\"\"&\"\"\"\"$V#*$)F& \"\"%F(\"$5)*$)F&\"\"$F(\"%!3\"*$)F&\"\"#F(\"$?(F&\"$S#\"#KF(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$,.*$)%\"xG\"\"&\"\"\"\"$V#*$ )F)\"\"%F+\"$5)*$)F)\"\"$F+\"%!3\"*$)F)\"\"#F+\"$?(F)\"$S#\"#KF+F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,,*$)%\"xG\"\"%\"\"\"\"%:7*$)F&\"\"$F( \"%SK*$)F&\"\"#F(F-F&\"%S9\"$S#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, $*$),&%\"xG\"\"$\"\"#\"\"\"\"\"%F*\"#:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 9 "Example 3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 0 "" {TEXT 258 8 "Question" }{TEXT 291 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 7 "Given " }{XPPEDIT 18 0 "y = (x^3+7*x-2)^7 ;" "6#/%\"yG*$,(*$%\"xG\"\"$\"\"\"*&\"\"(F*F(F*F*\"\"#!\"\"F," }{TEXT -1 7 ", find " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 269 8 "Solution" }{TEXT 292 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "u = x^3+7*x-2;" "6#/%\"uG,(*$%\"xG\"\"$\"\"\"*& \"\"(F)F'F)F)\"\"#!\"\"" }{TEXT -1 11 ", so that " }{XPPEDIT 18 0 "y \+ = u^7;" "6#/%\"yG*$%\"uG\"\"(" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 8 "We have " }{XPPEDIT 18 0 "du/dx = 3*x^2+7;" "6#/*&%#duG\" \"\"%#dxG!\"\",&*&\"\"$F&*$%\"xG\"\"#F&F&\"\"(F&" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "dy/du = 7*u^6;" "6#/*&%#dyG\"\"\"%#duG!\"\"*&\"\"(F&*$ %\"uG\"\"'F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 24 "Then, by \+ the chain rule," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy /dx = dy/du" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&F%F&%#duGF(" }{TEXT -1 2 " \+ " }{XPPEDIT 18 0 "du/dx = 7*u^6*`.`*(3*x^2+7);" "6#/*&%#duG\"\"\"%#dxG !\"\"**\"\"(F&*$%\"uG\"\"'F&%\".GF&,&*&\"\"$F&*$%\"xG\"\"#F&F&F*F&F&" }}{PARA 256 "" 0 "" {TEXT -1 4 " = " }{XPPEDIT 18 0 "7*(x^3+7*x-2)^6* (3*x^2+7);" "6#*(\"\"(\"\"\"*$,(*$%\"xG\"\"$F%*&F$F%F)F%F%\"\"#!\"\"\" \"'F%,&*&F*F%*$F)F,F%F%F$F%F%" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 293 13 "Alternatively" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 7*(x^3+7*x-2)^6;" "6#/*&%#dyG\" \"\"%#dxG!\"\"*&\"\"(F&*$,(*$%\"xG\"\"$F&*&F*F&F.F&F&\"\"#F(\"\"'F&" } {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "[x^3+7*x-2];" "6#7#,(*$%\"xG\"\"$\"\"\" *&\"\"(F(F&F(F(\"\"#!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = 7*(x^3+7*x-2)^6*(3*x^2+7);" "6#/%!G*(\" \"(\"\"\"*$,(*$%\"xG\"\"$F'*&F&F'F+F'F'\"\"#!\"\"\"\"'F',&*&F,F'*$F+F. F'F'F&F'F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Diff((x^3+7*x-2)^7,x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$*$),(*$)%\"xG\"\"$\"\"\"F- F+\"\"(!\"#F-F.F-F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&),(*$)%\"xG \"\"$\"\"\"F+F)\"\"(!\"#F+\"\"'F+,&*$)F)\"\"#F+F*F,F+F+F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "Note that we obta in the same result, if we expand the expression before differentiating ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "(x^3+7*x-2)^7;\nexpand(%);\nDiff(%,x);\nvalue(%);\nfa ctor(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*$),(*$)%\"xG\"\"$\"\"\"F* F(\"\"(!\"#F*F+F*" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,L!$G\"\"\"\"*$)% \"xG\"\")F%!(3T\\\"*$)F(\"#:F%\"&*37*$)F(\"#8F%\"&vp)*$)F(\"#6F%\"'2TR *$)F(\"\"$F%\"'GD>*$)F(\"#9F%!&!H5*$)F(\"#;F%!$)e*$)F(\"#=F%!#9*$)F(\" #@F%F%*$)F(\"#>F%\"#\\*$)F(\"#.#" }}{PARA 12 "" 1 " " {XPPMATH 20 "6#-%%DiffG6$,L!$G\"\"\"\"*$)%\"xG\"\")F(!(3T\\\"*$)F+\" #:F(\"&*37*$)F+\"#8F(\"&vp)*$)F+\"#6F(\"'2TR*$)F+\"\"$F(\"'GD>*$)F+\"# 9F(!&!H5*$)F+\"#;F(!$)e*$)F+\"#=F(!#9*$)F+\"#@F(F(*$)F+\"#>F(\"#\\*$)F +\"#.#F+" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,J*$)% \"xG\"\"(\"\"\"!)kG&>\"*$)F&\"#9F(\"'N8=*$)F&\"#7F(\"(v18\"*$)F&\"#5F( \"(x^L%*$)F&\"\"#F(\"'%ex&*$)F&\"#8F(!'gS9*$)F&\"#:F(!%3%**$)F&\"#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&),(*$)%\"xG\"\"$\"\"\"F+F)\"\"(!\"#F+\"\"'F +,&*$)F)\"\"#F+F*F,F+F+F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{PARA 257 "" 0 "" {TEXT 258 8 "Question" }{TEXT 294 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 7 "Given " }{XPPEDIT 18 0 "y = 1/(1+x^2);" "6#/%\" yG*&\"\"\"F&,&F&F&*$%\"xG\"\"#F&!\"\"" }{TEXT -1 7 ", find " } {XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 269 8 "Solutio n" }{TEXT 295 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "u = 1+x^2;" "6#/%\"uG,&\"\"\"F&*$%\"xG\"\"#F&" }{TEXT -1 11 ", so t hat " }{XPPEDIT 18 0 "y = u^(-1);" "6#/%\"yG)%\"uG,$\"\"\"!\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 8 "We have " }{XPPEDIT 18 0 "du/dx = 2*x;" "6#/*&%#duG\"\"\"%#dxG!\"\"*&\"\"#F&%\"xGF&" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "dy/du = -u^(-2);" "6#/*&%#dyG\"\"\"%#duG!\" \",$)%\"uG,$\"\"#F(F(" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "-1/u^2" "6#,$ *&\"\"\"F%*$%\"uG\"\"#!\"\"F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 24 "Then, by the chain rule," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = dy/du" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&F%F&%# duGF(" }{TEXT -1 2 " " }{XPPEDIT 18 0 "du/dx = -1/(u^2);" "6#/*&%#duG \"\"\"%#dxG!\"\",$*&F&F&*$%\"uG\"\"#F(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "`.`*2*x;" "6#*(%\".G\"\"\"\"\"#F%%\"xGF%" }{TEXT -1 2 " " }} {PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = -2*x/((1+x^2)^2 );" "6#/%!G,$*(\"\"#\"\"\"%\"xGF(*$,&F(F(*$F)F'F(F'!\"\"F-" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 296 13 "Alternatively" }{TEXT -1 8 ", si nce " }{XPPEDIT 18 0 "y = (1+x^2)^(-1);" "6#/%\"yG),&\"\"\"F'*$%\"xG\" \"#F',$F'!\"\"" }{TEXT -1 1 "," }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx = (-1)*(1+x^2)^(-2);" "6#/*&%#dyG\"\"\"%#dxG!\"\" *&,$F&F(F&),&F&F&*$%\"xG\"\"#F&,$F/F(F&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[ 1+x^2];" "6#7#,&\"\"\"F%*$%\"xG\"\"#F%" }{TEXT -1 1 " " }}{PARA 256 " " 0 "" {TEXT -1 4 " = " }{XPPEDIT 18 0 "(-1)*(1+x^2)^(-2)*2*x;" "6#** ,$\"\"\"!\"\"F%),&F%F%*$%\"xG\"\"#F%,$F+F&F%F+F%F*F%" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 4 " = " }{XPPEDIT 18 0 "-2*x/((1+x^2)^2) ;" "6#,$*(\"\"#\"\"\"%\"xGF&*$,&F&F&*$F'F%F&F%!\"\"F+" }{TEXT -1 1 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Diff(1/(1+x^2),x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$*&\"\"\"F',&*$)%\"xG\"\"#F'F'F'F'!\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%\"xG\"\"\"*$),&*$)F%\"\"#F&F&F&F&F,F&!\"\"!\" #" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 5" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 0 "" {TEXT 258 8 "Question" }{TEXT 371 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 7 "Given " }{XPPEDIT 18 0 "y = 1/sqrt(x^3-2*x);" "6#/%\"yG*&\"\"\"F&-%% sqrtG6#,&*$%\"xG\"\"$F&*&\"\"#F&F,F&!\"\"F0" }{TEXT -1 7 ", find " } {XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 269 8 "Solutio n" }{TEXT 372 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "u = x^3-2*x;" "6#/%\"uG,&*$%\"xG\"\"$\"\"\"*&\"\"#F)F'F)!\"\"" } {TEXT -1 11 ", so that " }{XPPEDIT 18 0 "y = u^(-1/2);" "6#/%\"yG)%\" uG,$*&\"\"\"F)\"\"#!\"\"F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 8 "We have " }{XPPEDIT 18 0 "du/dx = 3*x^2-2;" "6#/*&%#duG\"\"\"%#d xG!\"\",&*&\"\"$F&*$%\"xG\"\"#F&F&F.F(" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "dy/du = -1/2;" "6#/*&%#dyG\"\"\"%#duG!\"\",$*&F&F&\"\"#F(F(" } {TEXT -1 1 " " }{XPPEDIT 18 0 "u^(-3/2)" "6#)%\"uG,$*&\"\"$\"\"\"\"\"# !\"\"F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 24 "Then, by the c hain rule," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = dy/du" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&F%F&%#duGF(" }{TEXT -1 2 " " } {XPPEDIT 18 0 "du/dx = -1/2;" "6#/*&%#duG\"\"\"%#dxG!\"\",$*&F&F&\"\"# F(F(" }{TEXT -1 2 " " }{XPPEDIT 18 0 "u^(-3/2)*`.`*(3*x^2-2);" "6#*() %\"uG,$*&\"\"$\"\"\"\"\"#!\"\"F+F)%\".GF),&*&F(F)*$%\"xGF*F)F)F*F+F)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` \+ = -(3*x^2-2)/(2*(x^3-2*x)^(3/2));" "6#/%!G,$*&,&*&\"\"$\"\"\"*$%\"xG\" \"#F*F*F-!\"\"F**&F-F*),&*$F,F)F**&F-F*F,F*F.*&F)F*F-F.F*F.F." }{TEXT -1 2 ". " }{TEXT 373 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Alternatively , let " }{XPPEDIT 18 0 "f(x)=1/sqrt(x^3-2*x)" "6#/-%\"fG6#%\"xG*&\"\" \"F)-%%sqrtG6#,&*$F'\"\"$F)*&\"\"#F)F'F)!\"\"F2" }{TEXT -1 7 ". Then \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = -1/2;" "6 #/*&%#dyG\"\"\"%#dxG!\"\",$*&F&F&\"\"#F(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(x^3-2*x)^(-3/2)" "6#),&*$%\"xG\"\"$\"\"\"*&\"\"#F(F&F(!\"\",$*& F'F(F*F+F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dx G!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[ x^3-2*x]" "6#7#,&*$%\"xG\"\" $\"\"\"*&\"\"#F(F&F(!\"\"" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=-1/2" "6#/%!G,$*&\"\"\"F'\"\"#!\"\"F)" } {TEXT -1 1 " " }{XPPEDIT 18 0 "(x^3-2*x)^(-3/2)*(3*x^2-2)" "6#*&),&*$% \"xG\"\"$\"\"\"*&\"\"#F)F'F)!\"\",$*&F(F)F+F,F,F),&*&F(F)*$F'F+F)F)F+F ,F)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=-(3*x^2-2)/(2*(x^3-2*x)^(3/2))" "6#/%!G,$*&,&*&\"\"$\"\"\"*$%\"x G\"\"#F*F*F-!\"\"F**&F-F*),&*$F,F)F**&F-F*F,F*F.*&F)F*F-F.F*F.F." } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Diff(1/sqrt(x^3-2*x),x);\nvalue(%);\nnormal(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$*&\"\"\"F'*$,&*$)%\"xG\" \"$F'F'*&\"\"#F'F,F'!\"\"#F'F/F0F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,$*(\"\"#!\"\",&*$)%\"xG\"\"$\"\"\"F,*&F%F,F*F,F&#!\"$F%,&*&F+F,)F*F%F ,F,F%F&F,F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\"#!\"\",&*$)%\"x G\"\"$\"\"\"F,*&F%F,F*F,F&#!\"$F%,&*&F+F,)F*F%F,F,F%F&F,F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 6" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 0 "" {TEXT 258 8 "Question " }{TEXT 337 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 7 "Given " }{XPPEDIT 18 0 "y = ``[3];" "6#/%\"yG&%!G6#\"\"$" }{XPPEDIT 18 0 "sqrt(1-8*x)" " 6#-%%sqrtG6#,&\"\"\"F'*&\"\")F'%\"xGF'!\"\"" }{TEXT -1 7 ", find " } {XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 269 8 "Solutio n" }{TEXT 338 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "u = 1-8*x;" "6#/%\"uG,&\"\"\"F&*&\"\")F&%\"xGF&!\"\"" }{TEXT -1 11 ", so that " }{XPPEDIT 18 0 "y = u^(1/3);" "6#/%\"yG)%\"uG*&\"\"\"F( \"\"$!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 8 "We have " } {XPPEDIT 18 0 "du/dx = -8;" "6#/*&%#duG\"\"\"%#dxG!\"\",$\"\")F(" } {TEXT -1 6 " and " }{XPPEDIT 18 0 "dy/du = 1/3;" "6#/*&%#dyG\"\"\"%#d uG!\"\"*&F&F&\"\"$F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "u^(-2/3);" "6#)% \"uG,$*&\"\"#\"\"\"\"\"$!\"\"F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 24 "Then, by the chain rule," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = dy/du" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&F%F&%# duGF(" }{TEXT -1 2 " " }{XPPEDIT 18 0 "du/dx = 1/3;" "6#/*&%#duG\"\" \"%#dxG!\"\"*&F&F&\"\"$F(" }{TEXT -1 2 " " }{XPPEDIT 18 0 "u^(-2/3)*` .`*(-8);" "6#*()%\"uG,$*&\"\"#\"\"\"\"\"$!\"\"F+F)%\".GF),$\"\")F+F)" }{XPPEDIT 18 0 "`` = -8/3;" "6#/%!G,$*&\"\")\"\"\"\"\"$!\"\"F*" } {TEXT -1 2 " " }{XPPEDIT 18 0 "u^(-2/3);" "6#)%\"uG,$*&\"\"#\"\"\"\" \"$!\"\"F*" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`` = -8/3;" "6#/%!G,$*&\"\")\"\"\"\"\"$!\"\"F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(1-8*x)^(-2/3);" "6#),&\"\"\"F%*&\"\")F%%\"xG F%!\"\",$*&\"\"#F%\"\"$F)F)" }{TEXT -1 2 ". " }{TEXT 339 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Alternatively, since " }{XPPEDIT 18 0 "y = (1-8 *x)^(1/3);" "6#/%\"yG),&\"\"\"F'*&\"\")F'%\"xGF'!\"\"*&F'F'\"\"$F+" } {TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/ dx = 1/3;" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&F&F&\"\"$F(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "(1-8*x)^(-2/3);" "6#),&\"\"\"F%*&\"\")F%%\"xGF%!\"\",$* &\"\"#F%\"\"$F)F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\" \"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[1-8*x];" "6#7#,&\"\" \"F%*&\"\")F%%\"xGF%!\"\"" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1/3;" "6#/%!G*&\"\"\"F&\"\"$!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "(1-8*x)^(-2/3)*(-8);" "6#*&),&\"\"\"F&* &\"\")F&%\"xGF&!\"\",$*&\"\"#F&\"\"$F*F*F&,$F(F*F&" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = -8/3;" "6#/%!G,$ *&\"\")\"\"\"\"\"$!\"\"F*" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(1-8*x)^(-2 /3);" "6#),&\"\"\"F%*&\"\")F%%\"xGF%!\"\",$*&\"\"#F%\"\"$F)F)" }{TEXT -1 2 ". " }{TEXT 374 0 "" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Diff((1-8*x)^(1/3),x) ;\nvalue(%);\nnormal(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$ *$),&\"\"\"F)*&\"\")F)%\"xGF)!\"\"#F)\"\"$F)F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\")\"\"\"\"\"$!\"\",&F&F&*&F%F&%\"xGF&F(#!\"#F'F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\"\")\"\"\"\"\"$!\"\",&F&F&*&F %F&%\"xGF&F(#!\"#F'F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 7" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{PARA 257 "" 0 "" {TEXT 258 8 "Question" }{TEXT 356 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 7 "Given " }{XPPEDIT 18 0 "y = 1/(sqrt(1+x^2)-x^2) ;" "6#/%\"yG*&\"\"\"F&,&-%%sqrtG6#,&F&F&*$%\"xG\"\"#F&F&*$F-F.!\"\"F0 " }{TEXT -1 7 ", find " }{XPPEDIT 18 0 "dy/dx" "6#*&%#dyG\"\"\"%#dxG! \"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 269 8 "Solution" }{TEXT 357 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 17 "First note that " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG! \"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[sqrt(1+x^2)] = 1/2;" "6#/7#-%%s qrtG6#,&\"\"\"F)*$%\"xG\"\"#F)*&F)F)F,!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "(1+x^2)^(-1/2);" "6#),&\"\"\"F%*$%\"xG\"\"#F%,$*&F%F%F( !\"\"F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG! \"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[1+x^2] = 1/2;" "6#/7#,&\"\"\"F& *$%\"xG\"\"#F&*&F&F&F)!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(1+x^2)^( -1/2)*2*x = x/sqrt(1+x^2);" "6#/*(),&\"\"\"F'*$%\"xG\"\"#F',$*&F'F'F*! \"\"F-F'F*F'F)F'*&F)F'-%%sqrtG6#,&F'F'*$F)F*F'F-" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "y = (sqrt(1+x^2)-x ^2)^(-1);" "6#/%\"yG),&-%%sqrtG6#,&\"\"\"F+*$%\"xG\"\"#F+F+*$F-F.!\"\" ,$F+F0" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "dy/dx = (-1)*(sqrt(1+x^2)-x^2)^(-2);" "6#/*&%#dyG\"\"\" %#dxG!\"\"*&,$F&F(F&),&-%%sqrtG6#,&F&F&*$%\"xG\"\"#F&F&*$F2F3F(,$F3F(F &" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "[sqrt(1+x^2)-x^2];" "6#7#,&-%%sqrtG6#,& \"\"\"F)*$%\"xG\"\"#F)F)*$F+F,!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "` ` = (-1)*(x^2+sqrt(1+x^2))^(-2);" "6#/%!G*&,$\"\"\"!\"\"F'),&*$%\"xG\" \"#F'-%%sqrtG6#,&F'F'*$F,F-F'F',$F-F(F'" }{XPPEDIT 18 0 "(1/sqrt(1+x^2 )-2)*x;" "6#*&,&*&\"\"\"F&-%%sqrtG6#,&F&F&*$%\"xG\"\"#F&!\"\"F&F-F.F&F ,F&" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (-1)*(x^2+sqrt(1+x^2))^(-2);" "6#/%!G*&,$\" \"\"!\"\"F'),&*$%\"xG\"\"#F'-%%sqrtG6#,&F'F'*$F,F-F'F',$F-F(F'" } {XPPEDIT 18 0 "``((1-2*sqrt(1+x^2))/sqrt(x^2+1))*x;" "6#*&-%!G6#*&,&\" \"\"F)*&\"\"#F)-%%sqrtG6#,&F)F)*$%\"xGF+F)F)!\"\"F)-F-6#,&*$F1F+F)F)F) F2F)F1F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = x*(2*sqrt(1+x^2)-1)/((x^2- sqrt(1+x^2))^2*sqrt(1+x^2));" "6#/%!G*(%\"xG\"\"\",&*&\"\"#F'-%%sqrtG6 #,&F'F'*$F&F*F'F'F'F'!\"\"F'*&,&*$F&F*F'-F,6#,&F'F'*$F&F*F'F0F*-F,6#,& F'F'*$F&F*F'F'F0" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "f := x -> 1/(sqrt(1+x^2)-x^2 ):\n'f(x)'=f(x);\nDiff('f(x)',x)=normal(diff(f(x),x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&\"\"\"F),&*$,&F)F)*$)F'\"\"#F)F)#F )F/F)F-!\"\"F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"fG6#% \"xGF***F*\"\"\",&F,!\"\"*&\"\"#F,,&F,F,*$)F*F0F,F,#F,F0F,F,,&*$F1F4F. F2F,!\"#F1#F.F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "The denom inator " }{XPPEDIT 18 0 "x^2-sqrt(1+x^2)" "6#,&*$%\"xG\"\"#\"\"\"-%%sq rtG6#,&F'F'*$F%F&F'!\"\"" }{TEXT -1 23 " of the expression for " } {XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 21 " is equal to 0 wh en " }{XPPEDIT 18 0 "x =``" "6#/%\"xG%!G" }{TEXT 366 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 " sqrt(2+2*sqrt(5))/2" "6#*&-%%sqrtG6#,&\"\"#\"\" \"*&F(F)-F%6#\"\"&F)F)F)F(!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 7 "Indeed " }{XPPEDIT 18 0 "x^2-sqrt(1+x^2)=0" "6#/,&*$%\"xG \"\"#\"\"\"-%%sqrtG6#,&F(F(*$F&F'F(!\"\"\"\"!" }{TEXT -1 6 " when " } {XPPEDIT 18 0 "x^2=sqrt(1+x^2)" "6#/*$%\"xG\"\"#-%%sqrtG6#,&\"\"\"F+*$ F%F&F+" }{TEXT -1 36 ". Squaring this last equation gives " }{XPPEDIT 18 0 "x^4=1+x^2" "6#/*$%\"xG\"\"%,&\"\"\"F(*$F%\"\"#F(" }{TEXT -1 34 " or, after completing the square, " }{XPPEDIT 18 0 "x^4-x^2+1/4=5/4" " 6#/,(*$%\"xG\"\"%\"\"\"*$F&\"\"#!\"\"*&F(F(F'F+F(*&\"\"&F(F'F+" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 38 "The left side of the las t equation is " }{XPPEDIT 18 0 "(x^2-1/2)^2 " "6#*$,&*$%\"xG\"\"#\"\" \"*&F(F(F'!\"\"F*F'" }{TEXT -1 15 ", so we obtain " }{XPPEDIT 18 0 "x^ 2-1/2 =`` " "6#/,&*$%\"xG\"\"#\"\"\"*&F(F(F'!\"\"F*%!G" }{TEXT 367 1 " +" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(5)/2" "6#*&-%%sqrtG6#\"\"&\"\" \"\"\"#!\"\"" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "x^2=1/2" "6#/*$%\"xG \"\"#*&\"\"\"F(F&!\"\"" }{TEXT -1 1 " " }{TEXT 370 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(5)/2" "6#*&-%%sqrtG6#\"\"&\"\"\"\"\"#!\"\"" } {TEXT -1 54 ". Taking the minus sign does not give real values for " } {TEXT 369 1 "x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 27 "Taking the plus sign gives " }{XPPEDIT 18 0 "x^2=(2 + 2*sqrt(5))/4" "6#/*$% \"xG\"\"#*&,&F&\"\"\"*&F&F)-%%sqrtG6#\"\"&F)F)F)\"\"%!\"\"" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "x =``" "6#/%\"xG%!G" }{TEXT 368 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 " sqrt(2+2*sqrt(5))/2" "6#*&-%%sqrtG6#, &\"\"#\"\"\"*&F(F)-F%6#\"\"&F)F)F)F(!\"\"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 "This means that the graph of " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" } {TEXT -1 38 " has the vertical asymptotes given by " }{XPPEDIT 18 0 "x =``" "6#/%\"xG%!G" }{TEXT 365 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 " \+ sqrt(2+2*sqrt(5))/2" "6#*&-%%sqrtG6#,&\"\"#\"\"\"*&F(F)-F%6#\"\"&F)F)F )F(!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "solve(sqrt(1+x^2)=x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$*&\"\"#!\"\",&F%\"\"\"*&F%F(\"\"&#F(F%F(F+ F&,$*&F%F&F'F+F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "The following picture shows the graphs of " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 5 " (in " }{TEXT 268 3 "r ed" }{TEXT -1 6 ") and " }{XPPEDIT 18 0 "y = `f '`(x);" "6#/%\"yG-%$f~ 'G6#%\"xG" }{TEXT -1 5 " (in " }{TEXT 256 4 "blue" }{TEXT -1 3 "). " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 261 "f := x -> 1/(sqrt(1+x^2)-x^2):\n'f(x)'=f(x);\ncc := evalf(1/2*s qrt(2+2*sqrt(5))):\np1 := plot([f(x),D(f)(x)],x=-3..3,y=-3..3,color=[r ed,blue],discont=true):\np2 := plot([[[-cc,-3.5],[-cc,3.5]],[[cc,-3.5] ,[cc,3.5]]],color=black,linestyle=3):\nplots[display]([p1,p2]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*&\"\"\"F),&*$,&F)F)*$)F '\"\"#F)F)#F)F/F)F-!\"\"F1" }}{PARA 13 "" 1 "" {GLPLOT2D 476 476 476 {PLOTDATA 2 "6*-%'CURVESG6&7gn7$$!\"$\"\"!$!3FU[a`o*Hr\"!#=7$$!3Ci[/Y \\LiH!#<$!3sw:'4jo-x\"F-7$$!3HYC*Ryi&HHF1$!3rOto$)\\bA=F-7$$!3qs5Bgqq# *GF1$!3%\\Mhv'pF%)=F-7$$!35&o3G.2c&GF1$!3%)>6loQv\\>F-7$$!3i@0MNLo=GF1 $!35^cU]a]=?F-7$$!3!)e_AT.X%y#F1$!3KixEmWo&3#F-7$$!3Q48neS+\\FF1$!3Y5I S'=&**e@F-7$$!3!QeZDYXBr#F1$!3gQ)f&z#R\"RAF-7$$!3SXl+JW!en#F1$!3:->HG+ wBBF-7$$!33zGg)[<#QEF1$!30;vuTIDF1$!3'fN%e*o$z:FF-7$$!3#*[ q&RE`V\\#F1$!3:00\\M#)GHGF-7$$!3)Gzp^].;Y#F1$!3kHqa[p**QHF-7$$!3IdsMu3 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\\Yq&GF-7$Fdbn$\"319`f<>C&o#F-7$Fibn$\"3hd8-[s(3`#F-7$F^cn$\"3gSmln-># Q#F-7$Fccn$\"3SUK$oL?-D#F-7$Fhcn$\"3*zKP1BGb7#F-7$F]dn$\"3@KD@nH86?F-7 $Fbdn$\"3F-7$Fgdn$\"3#)4QvRDG4=F-7$F\\en$\"3#*Q()y3!zEs\"F-7 $Faen$\"30-QTF/_O;F-7$Ffen$\"3a[juDY_f:F-7$F[fn$\"3]ZuuosB#[\"F--F^fn6 &F`fnFdfnFdfnFafn-%'POINTSG6%7$$!+]'>?F\"Fd^l$\"\"\"\"\"*7$$\"+]'>?F\" Fd^lFgjpF]fn-Fbjp6%7$Fejp$!+Az)yv\"Fijp7$F[[q$\"+Az)yv\"FijpF_jp-F$6%7 $7$$!31+++]'>?F\"F1$!3++++++++NF17$Fi[q$\"3++++++++NF1-F^fn6&F`fnF*F*F *-%*LINESTYLEG6#F\\fn-F$6%7$7$$\"31+++]'>?F\"F1F[\\q7$Fi\\qF^\\qF`\\qF b\\q-%+AXESLABELSG6%Q\"x6\"Q\"yF`]q-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(F[ fnFi]q" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 8" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 0 "" {TEXT 258 8 "Question" } {TEXT 378 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }{XPPEDIT 18 0 "g(x) = x*(x-1);" "6#/-%\"gG6#%\"xG*&F'\"\"\",&F'F)F)!\"\"F)" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "f(x)=g(g(g(x)))" "6#/-%\"fG6#%\"xG-%\"gG6 #-F)6#-F)6#F'" }{TEXT -1 7 ", find " }{XPPEDIT 18 0 "`f '`(x) = d/dx; " "6#/-%$f~'G6#%\"xG*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "[g(g(g(x)))]" "6#7#-%\"gG6#-F%6#-F%6#%\"xG" }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT 269 8 "Solution" }{TEXT 379 2 ": " } {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "g( x) = x^2-x;" "6#/-%\"gG6#%\"xG,&*$F'\"\"#\"\"\"F'!\"\"" }{TEXT -1 18 " , it follows that " }{XPPEDIT 18 0 "g*`'`(x) = 2*x-1;" "6#/*&%\"gG\"\" \"-%\"'G6#%\"xGF&,&*&\"\"#F&F*F&F&F&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "u(x)=g(g(x))" "6#/-%\"uG6#%\"xG-%\"gG6#-F)6#F'" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 25 "Then, by the chain rule, " }{XPPEDIT 18 0 "`u '`(x) = `g '`(g(x))*`g '`(x);" "6#/-%$u~'G6#%\"xG*&-%$g~'G6#- %\"gG6#F'\"\"\"-F*6#F'F/" }{TEXT -1 9 ", so that" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`u '`(x) = (2*(x^2-x)-1)*(2*x-1);" "6#/ -%$u~'G6#%\"xG*&,&*&\"\"#\"\"\",&*$F'F+F,F'!\"\"F,F,F,F/F,,&*&F+F,F'F, F,F,F/F," }{XPPEDIT 18 0 "``=(2*x^2-2*x-1)*(2*x-1)" "6#/%!G*&,(*&\"\"# \"\"\"*$%\"xGF(F)F)*&F(F)F+F)!\"\"F)F-F),&*&F(F)F+F)F)F)F-F)" } {XPPEDIT 18 0 "``=4*x^3-6*x^2+1" "6#/%!G,(*&\"\"%\"\"\"*$%\"xG\"\"$F(F (*&\"\"'F(*$F*\"\"#F(!\"\"F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Alternatively, since " } {XPPEDIT 18 0 "u(x)=g(x)^2-g(x)" "6#/-%\"uG6#%\"xG,&*$-%\"gG6#F'\"\"# \"\"\"-F+6#F'!\"\"" }{XPPEDIT 18 0 "``=(x^2-x)^2-(x^2-x)" "6#/%!G,&*$, &*$%\"xG\"\"#\"\"\"F)!\"\"F*F+,&*$F)F*F+F)F,F," }{XPPEDIT 18 0 "``=x^4 -2*x^3+x^2-x^2+x" "6#/%!G,,*$%\"xG\"\"%\"\"\"*&\"\"#F)*$F'\"\"$F)!\"\" *$F'F+F)*$F'F+F.F'F)" }{XPPEDIT 18 0 "``=x^4-2*x^3+x" "6#/%!G,(*$%\"xG \"\"%\"\"\"*&\"\"#F)*$F'\"\"$F)!\"\"F'F)" }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 16 "it follows that " }{XPPEDIT 18 0 "`u '`(x) = 4*x^3 -6*x^2+1;" "6#/-%$u~'G6#%\"xG,(*&\"\"%\"\"\"*$F'\"\"$F+F+*&\"\"'F+*$F' \"\"#F+!\"\"F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "Now " }{XPPEDIT 18 0 "f(x)=g(u(x))" "6#/-% \"fG6#%\"xG-%\"gG6#-%\"uG6#F'" }{TEXT -1 24 " so, by the chain rule, \+ " }{XPPEDIT 18 0 "`f '`(x) = `g '`(u(x))*`u '`(x);" "6#/-%$f~'G6#%\"xG *&-%$g~'G6#-%\"uG6#F'\"\"\"-%$u~'G6#F'F/" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 7 "Hence " }{XPPEDIT 18 0 "`f '`(x) = (2*u(x)-1)*`u ' `(x);" "6#/-%$f~'G6#%\"xG*&,&*&\"\"#\"\"\"-%\"uG6#F'F,F,F,!\"\"F,-%$u~ 'G6#F'F," }{XPPEDIT 18 0 " ``=``(2*x^4-4*x^3+2*x-1)*(2*x^2-2*x-1)*(2*x -1)" "6#/%!G*(-F$6#,**&\"\"#\"\"\"*$%\"xG\"\"%F+F+*&F.F+*$F-\"\"$F+!\" \"*&F*F+F-F+F+F+F2F+,(*&F*F+*$F-F*F+F+*&F*F+F-F+F2F+F2F+,&*&F*F+F-F+F+ F+F2F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "g := x -> x*(x-1);\nexpand(g(g(g(x) )));\nfactor(%);\nDiff(%,x);\nvalue(%);\nexpand(%);\nfactor(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrow GF(*&9$\"\"\",&F-F.F.!\"\"F.F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# ,2*$)%\"xG\"\")\"\"\"F(*&\"\"%F()F&\"\"(F(!\"\"*&\"\"#F()F&\"\"&F(F(*& F1F()F&F*F(F-*&F*F()F&\"\"'F(F(*&F/F()F&\"\"$F(F(*$)F&F/F(F(F&F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#**%\"xG\"\"\",&F$F%F%!\"\"F%,(*$)F$\" \"#F%F%F$F'F%F'F%,**$)F$\"\"%F%F%*&F+F%)F$\"\"$F%F'F$F%F%F'F%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$**%\"xG\"\"\",&F'F(F(!\"\"F( ,(*$)F'\"\"#F(F(F'F*F(F*F(,**$)F'\"\"%F(F(*&F.F()F'\"\"$F(F*F'F(F(F*F( F'" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,**(,&%\"xG\"\"\"F'!\"\"F',(*$)F &\"\"#F'F'F&F(F'F(F',**$)F&\"\"%F'F'*&F,F')F&\"\"$F'F(F&F'F'F(F'F'*(F& F'F)F'F-F'F'**F&F'F%F',&*&F,F'F&F'F'F'F(F'F-F'F'**F&F'F%F'F)F',(*&F0F' F2F'F'*&\"\"'F'F+F'F(F'F'F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,2\" \"\"!\"\"*&\"\"#F$%\"xGF$F$*&\"#?F$)F(\"\"$F$F%*&\"\"'F$)F(F'F$F$*&\"# 5F$)F(\"\"%F$F$*&\"\")F$)F(\"\"(F$F$*&\"#GF$)F(F.F$F%*&\"#CF$)F(\"\"&F $F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*(,&*&\"\"#\"\"\"%\"xGF'F'F'!\" \"F',(*&F&F')F(F&F'F'*&F&F'F(F'F)F'F)F',**&F&F')F(\"\"%F'F'*&F1F')F(\" \"$F'F)*&F&F'F(F'F'F'F)F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "No te" }{TEXT -1 2 ": " }{XPPEDIT 18 0 "`f '`(x)=0" "6#/-%$f~'G6#%\"xG\" \"!" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "x=1/2" "6#/%\"xG*&\"\"\"F&\" \"#!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "(1+sqrt(3))/2" "6#*&,&\"\" \"F%-%%sqrtG6#\"\"$F%F%\"\"#!\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "(1 -sqrt(3))/2" "6#*&,&\"\"\"F%-%%sqrtG6#\"\"$!\"\"F%\"\"#F*" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "(1-sqrt(3+2*sqrt(3)))/2" "6#*&,&\"\"\"F%-%%sqrt G6#,&\"\"$F%*&\"\"#F%-F'6#F*F%F%!\"\"F%F,F/" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "(1+sqrt(3+2*sqrt(3)))/2" "6#*&,&\"\"\"F%-%%sqrtG6#,&\" \"$F%*&\"\"#F%-F'6#F*F%F%F%F%F,!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "g := x - > x*(x-1);\nsolve(diff(g(g(g(x))),x)):\nop(remove(_u->has(evalf(_u),Co mplex(1)),[%]));\nevalf(evalf[14](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&9$\"\"\",&F-F.F.!\"\" F.F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'#\"\"\"\"\"#,&F#F$*&F%!\" \"\"\"$F#F$,&F#F$*&F%F(F)F#F(,&F#F$*&F%F(,&F)F$*&F%F$F)F#F$F#F(,&F#F$* &F%F(F.F#F$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'$\"+++++]!#5$\"+/a-m8! \"*$!+QSDgOF%$!+%y)H7xF%$\"+y)H7x\"F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 14 "The graph of " }{XPPEDIT 18 0 "y=f(x )" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 50 " has two maximum points and t hree minimum points. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "g := x -> x*(x-1);\nplot(g(g(g(x))),x=-1. .2,y=-.3..0.5);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6\"6 $%)operatorG%&arrowGF(*&9$\"\"\",&F-F.F.!\"\"F.F(F(F(" }}{PARA 13 "" 1 "" {GLPLOT2D 603 304 304 {PLOTDATA 2 "6%-%'CURVESG6$7aq7$$!\"\"\"\"! $\"\"#F*7$$!3$***\\(=ng#=**!#=$\"3q=zo%H%*py\"!#<7$$!3()***\\PM@l$)*F0 $\"3i`Opr=())e\"F37$$!3!)**\\i:?ya(*F0$\"3t:Z)>21\\S\"F37$$!3u****\\(o UIn*F0$\"3,bR@lhRM7F37$$!3m**\\PfLI\"f*F0$\"3SAn5smmw5F37$$!3g***\\7.k &4&*F0$\"3.G4%H*pm5$*F07$$!3a**\\7.Z#yU*F0$\"37B]ZkGppzF07$$!3[*****\\ P&3Y$*F0$\"3XU*4dl2xt'F07$$!3R*\\7.ZVQ?*F0$\"3E)z$)4ODl$[F07$$!3I**\\i l:gh!*F0$\"3C**))ydk')=KF07$$!3K+v$4mf$>*)F0$\"35zw3/ahd=F07$$!3C++Dcx 6x()F0$\"3#[miK:-\\F(!#>7$$!3%)**\\Pff=d%)F0$!3w0^bC$)**)3\"F07$$!3b++ ]iTDP\")F0$!3_gU*fu@14#F07$$!35+]PM5uc!)F0$!3D**pO1(3)QAF07$$!3m***\\i !zAwzF0$!3#>HRyL*3^BF07$$!3K+]7yZr&*yF0$!3?l!oGry,V#F07$$!3*)******\\; ?:yF0$!3106[eimyCF07$$!3X**\\(=_)oMxF0$!3?dlqK(>!*\\#F07$$!37++v$RvTl( F0$!3]>6c:\"*e$\\#F07$$!3m**\\ilAmtvF0$!3CrOt:/hkCF07$$!3A****\\P\"\\J \\(F0$!3p:S4%R4UT#F07$$!3U**\\P%GFE<(F0$!3^DifkGeR?F07$$!3g***\\7V0@&o F0$!3k\"=:q!*GuY\"F07$$!3w++DcexdiF0$!3pI\"yU,fkt\"Feo7$$!3j***\\i+#QU cF0$\"3K&fRV\\5N9\"F07$$!3$****\\i!3%f+&F0$\"3XZ;W!fv$=AF07$$!31++vV2u )o%F0$\"3<]k?0AZ.EF07$$!3;++D\"oS:P%F0$\"3c@(R[Hb4)GF07$$!3.+voa5S3UF0 $\"3c;\"*=.w,#)HF07$$!3!***\\7G9EXSF0$\"3%Rc_*>P^bIF07$$!3w*\\i:!=7#)Q F0$\"3Fg'psU+B5$F07$$!3h*****\\<#)*=PF0$\"3$pSr$=vVBJF07$$!3x***\\7B&f JMF0$\"3tG+,nA6-JF07$$!3#*****\\(G3U9$F0$\"3Il\"oZ\\jF,$F07$$!3p***\\i ly1#GF0$\"3VjVTIDZTGF07$$!3Y*****\\-\\r\\#F0$\"3V(*H2(>@xg#F07$$!3?+++ vGVZ=F0$\"3E\"4_\\7y>+#F07$$!3_*****\\(4J@7F0$\"3:6hL.&=DK\"F07$$!3;,+ ]iIKFlFeo$\"3M_Vj\\;[))oFeo7$$\"3(R,++]siL#!#?$!3[$y#3w*)yIBFex7$$\"3K ,+++!R5'fFeo$!3y[=0P6[pbFeo7$$\"3!)***\\P/QBE\"F0$!3ceE9%pqY2\"F07$$\" 39******\\\"o?&=F0$!3/E&>'3s8N9F07$$\"3k++vVb4*\\#F0$!3%*4!o8Wf/t\"F07 $$\"3w++DJ'=_6$F0$!31HvKhAGE>F07$$\"3#4++vVy!ePF0$!3-:FL+$Gt0#F07$$\"3 '4+](=WU[VF0$!3)e0lr&3AC@F07$$\"3s****\\7B>&)\\F0$!33Uo5m^U[@F07$$\"3w ***\\P>:mk&F0$!3=%z%R_JfC@F07$$\"3d***\\iv&QAiF0$!3iwXs#[+.1#F07$$\"3j ++]PPBWoF0$!3&\\O/Y\"ofO>F07$$\"3%*)*****\\Nm'[(F0$!39q>A&yqet\"F07$$ \"36****\\(yb^6)F0$!3QuJ%)4S`_9F07$$\"3')***\\PMaKs)F0$!3K-2mA#pX3\"F0 7$$\"3a****\\7TW)R*F0$!3cRc-VX_;cFeo7$$\"3z*****\\@80+\"F3$\"3^\"[!Q*= \"yM^!#@7$$\"31++]7,Hl5F3$\"3Q?'**pVP.*oFeo7$$\"3()**\\P4w)R7\"F3$\"3, z1rPeCV8F07$$\"3;++]x%f\")=\"F3$\"3Y%Q=HYQr.#F07$$\"3!)**\\P/-a[7F3$\" 3SkIq?gA)f#F07$$\"3#***\\7Ly4!G\"F3$\"3&4#=ijD*)GGF07$$\"3/+](=Yb;J\"F 3$\"3v1$yr\\))3+$F07$$\"3+]7.P@3F8F3$\"37oD-lt(*fIF07$$\"3&**\\(=7)3DM \"F3$\"3V))*>$**3\"35$F07$$\"3#*\\PM([NzN\"F3$\"3F'Q:>s#3AJF07$$\"3')* ***\\i@Ot8F3$\"3(\\8h'RubAJF07$$\"3w\\(=ily% R,EF07$$\"3>+++!*>=+:F3$\"3]xDf>%eSA#F07$$\"3-++DE&4Qc\"F3$\"3C;u!3RK> :\"F07$$\"3=+]P%>5pi\"F3$!3o-L)pSg5*>Feo7$$\"39+++bJ*[o\"F3$!3#\\\\luP #4h9F07$$\"36+]7j17=& \\#F07$$\"31]7y![POx\"F3$!3+eKm.e'))\\#F07$$\"3/+]i]s1\"y\"F3$!3ubL4Fb `![#F07$$\"3/](o/-(\\)y\"F3$!3M$p8(GPOQCF07$$\"3-+DJ!zEfz\"F3$!3UcyjZ, UqBF07$$\"3,]i:glN.=F3$!3w]Y'\\n'ouAF07$$\"3++++Ijy5=F3$!3[qRu-Z0\\@F0 7$$\"3.+v=nIZU=F3$!31QRjW$3TA\"F07$$\"31+]P/)fT(=F3$\"3a)p@:$Hb&y%Feo7 $$\"3'**\\(oHRK*)=F3$\"3j=#[FR\"[L;F07$$\"32+++b!)[/>F3$\"3'*R4zKC0YIF 07$$\"3=+DJ!=_'>>F3$\"3?TPThvdYZF07$$\"31+]i0j\"[$>F3$\"3)z52Ig?xw'F07 $$\"3+v=UnU'H%>F3$\"3gI(QXd5$)*zF07$$\"3%*\\(=#HA6^>F3$\"3Z#)*)Q0)GtL* F07$$\"35Dc,\">g#f>F3$\"39TpNJf2z5F37$$\"3/+D\"G:3u'>F3$\"3KS)e8i\"[O7 F37$$\"3)\\P4Y6cb(>F3$\"3F*[YC+&f19F37$$\"39]iSwSq$)>F3$\"3utSq4a3!f\" F37$$\"31DJ?Q?&=*>F3$\"3i[)*>;rk(y\"F37$F+F+-%'COLOURG6&%$RGBG$\"#5F)$ F*F*Fb[m-%+AXESLABELSG6$Q\"x6\"Q\"yFg[m-%%VIEWG6$;F(F+;$!\"$F)$\"\"&F) " 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }} }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 13 "Example 9 ???" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 0 "" {TEXT 258 8 "Question" }{TEXT 375 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }{XPPEDIT 18 0 "g(x) = 4*x*(1-x);" "6#/-%\"gG6#%\"xG*(\"\"%\" \"\"F'F*,&F*F*F'!\"\"F*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "f(x)=g(g( g(x)))" "6#/-%\"fG6#%\"xG-%\"gG6#-F)6#-F)6#F'" }{TEXT -1 7 ", find " } {XPPEDIT 18 0 "`f '`(x) = d/dx;" "6#/-%$f~'G6#%\"xG*&%\"dG\"\"\"%#dxG! \"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[g(g(g(x)))]" "6#7#-%\"gG6#-F%6# -F%6#%\"xG" }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT 269 8 "Solution " }{TEXT 376 2 ": " }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "g(x) = 4*x-4*x^2;" "6#/-%\"gG6#%\"xG,&*&\"\"%\"\"\" F'F+F+*&F*F+*$F'\"\"#F+!\"\"" }{TEXT -1 18 ", it follows that " } {XPPEDIT 18 0 "`g '`(x) = 4-8*x;" "6#/-%$g~'G6#%\"xG,&\"\"%\"\"\"*&\" \")F*F'F*!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "u(x)=g(g(x))" "6#/-% \"uG6#%\"xG-%\"gG6#-F)6#F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 25 "Then, by the chain rule, " }{XPPEDIT 18 0 "`u '`(x) = `g '`(g(x ))*`g '`(x);" "6#/-%$u~'G6#%\"xG*&-%$g~'G6#-%\"gG6#F'\"\"\"-F*6#F'F/" }{TEXT -1 9 ", so that" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`u '`(x) = (4-8*(4*x-4*x^2))*(4-8*x);" "6#/-%$u~'G6#%\"xG*&,&\" \"%\"\"\"*&\"\")F+,&*&F*F+F'F+F+*&F*F+*$F'\"\"#F+!\"\"F+F3F+,&F*F+*&F- F+F'F+F3F+" }{XPPEDIT 18 0 "`` = (4-32*x+32*x^2)*(4-8*x);" "6#/%!G*&,( \"\"%\"\"\"*&\"#KF(%\"xGF(!\"\"*&F*F(*$F+\"\"#F(F(F(,&F'F(*&\"\")F(F+F (F,F(" }{XPPEDIT 18 0 "`` = 16-128*x+128*x^2-32*x+256*x^2-256*x^3;" "6 #/%!G,.\"#;\"\"\"*&\"$G\"F'%\"xGF'!\"\"*&F)F'*$F*\"\"#F'F'*&\"#KF'F*F' F+*&\"$c#F'*$F*F.F'F'*&F2F'*$F*\"\"$F'F+" }{TEXT -1 3 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=-256*x^3+384*x^2-160*x+1 6" "6#/%!G,**&\"$c#\"\"\"*$%\"xG\"\"$F(!\"\"*&\"$%QF(*$F*\"\"#F(F(*&\" $g\"F(F*F(F,\"#;F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 21 "Al ternatively, since " }{XPPEDIT 18 0 "u(x) = 4*g(x)-4*g(x)^2;" "6#/-%\" uG6#%\"xG,&*&\"\"%\"\"\"-%\"gG6#F'F+F+*&F*F+*$-F-6#F'\"\"#F+!\"\"" } {XPPEDIT 18 0 "`` = 4*(4*x-4*x^2)-4*(4*x-4*x^2)^2;" "6#/%!G,&*&\"\"%\" \"\",&*&F'F(%\"xGF(F(*&F'F(*$F+\"\"#F(!\"\"F(F(*&F'F(*$,&*&F'F(F+F(F(* &F'F(*$F+F.F(F/F.F(F/" }{XPPEDIT 18 0 "``=16*x-16*x^2-4*(16*x^2-32*x^3 +16*x^4)" "6#/%!G,(*&\"#;\"\"\"%\"xGF(F(*&F'F(*$F)\"\"#F(!\"\"*&\"\"%F (,(*&F'F(*$F)F,F(F(*&\"#KF(*$F)\"\"$F(F-*&F'F(*$F)F/F(F(F(F-" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= -64*x^ 4+128*x^3-80*x^2+16*x" "6#/%!G,**&\"#k\"\"\"*$%\"xG\"\"%F(!\"\"*&\"$G \"F(*$F*\"\"$F(F(*&\"#!)F(*$F*\"\"#F(F,*&\"#;F(F*F(F(" }{TEXT -1 2 ", \+ " }}{PARA 0 "" 0 "" {TEXT -1 16 "it follows that " }{XPPEDIT 18 0 "`u \+ '`(x) = -256*x^3+384*x^2-160*x+16;" "6#/-%$u~'G6#%\"xG,**&\"$c#\"\"\"* $F'\"\"$F+!\"\"*&\"$%QF+*$F'\"\"#F+F+*&\"$g\"F+F'F+F.\"#;F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Now " }{XPPEDIT 18 0 "f(x)=g(u(x))" "6#/-%\"fG6#%\"xG-%\"gG6#-%\"uG 6#F'" }{TEXT -1 24 " so, by the chain rule, " }{XPPEDIT 18 0 "`f '`(x) = `g '`(u(x))*`u '`(x);" "6#/-%$f~'G6#%\"xG*&-%$g~'G6#-%\"uG6#F'\"\" \"-%$u~'G6#F'F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x)=(4-8*u(x ))*`u '`(x)" "6#/-%$f~'G6#%\"xG*&,&\"\"%\"\"\"*&\"\")F+-%\"uG6#F'F+!\" \"F+-%$u~'G6#F'F+" }{XPPEDIT 18 0 " ``= (4-8*(-64*x^4+128*x^3-80*x^2+1 6*x))*(4-32*x+32*x^2)*(4-8*x)" "6#/%!G*(,&\"\"%\"\"\"*&\"\")F(,**&\"#k F(*$%\"xGF'F(!\"\"*&\"$G\"F(*$F/\"\"$F(F(*&\"#!)F(*$F/\"\"#F(F0*&\"#;F (F/F(F(F(F0F(,(F'F(*&\"#KF(F/F(F0*&F=F(*$F/F8F(F(F(,&F'F(*&F*F(F/F(F0F (" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(512*x^4-1024*x^3+640*x^2-128*x+4) *(4-32*x+32*x^2)*(4-8*x)" "6#/%!G*(,,*&\"$7&\"\"\"*$%\"xG\"\"%F)F)*&\" %C5F)*$F+\"\"$F)!\"\"*&\"$S'F)*$F+\"\"#F)F)*&\"$G\"F)F+F)F1F,F)F),(F,F )*&\"#KF)F+F)F1*&F:F)*$F+F5F)F)F),&F,F)*&\"\")F)F+F)F1F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "``=-131072*x^7+458752*x^6-638976*x^5+450560*x^4-1689 60*x^3+32256*x^2-2688*x+64" "6#/%!G,2*&\"'s58\"\"\"*$%\"xG\"\"(F(!\"\" *&\"'_(e%F(*$F*\"\"'F(F(*&\"'w*Q'F(*$F*\"\"&F(F,*&\"'g0XF(*$F*\"\"%F(F (*&\"'g*o\"F(*$F*\"\"$F(F,*&\"&cA$F(*$F*\"\"#F(F(*&\"%)o#F(F*F(F,\"#kF (" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 259 4 "Note" }{TEXT -1 6 ": " }{XPPEDIT 18 0 "f(x)=-16384*x^8+65536*x^7-106496*x^6+90112*x^5-42240*x^4+10752*x^3- 1344*x^2+ 64*x" "6#/-%\"fG6#%\"xG,2*&\"&%Q;\"\"\"*$F'\"\")F+!\"\"*&\"& Ob'F+*$F'\"\"(F+F+*&\"''\\1\"F+*$F'\"\"'F+F.*&\"&7,*F+*$F'\"\"&F+F+*& \"&SA%F+*$F'\"\"%F+F.*&\"&_2\"F+*$F'\"\"$F+F+*&\"%W8F+*$F'\"\"#F+F.*& \"#kF+F'F+F+" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 10 "The graph " }{XPPEDIT 18 0 "y=f(x)" "6#/% \"yG-%\"fG6#%\"xG" }{TEXT -1 45 " has 4 maximum points and 3 minimum p oints. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x) = \+ 0;" "6#/-%$f~'G6#%\"xG\"\"!" }{TEXT -1 6 " when " }{XPPEDIT 18 0 "x=1/ 2, 1/2+sqrt(2)/4, 1/2-sqrt(2)/4, 1/2-sqrt(2+sqrt(2))/4, 1/2+sqrt(2+sqr t(2))/4, 1/2-sqrt(2-sqrt(2))/4, 1/2+sqrt(2-sqrt(2))/4" "6)/%\"xG*&\"\" \"F&\"\"#!\"\",&*&F&F&F'F(F&*&-%%sqrtG6#F'F&\"\"%F(F&,&*&F&F&F'F(F&*&- F-6#F'F&F/F(F(,&*&F&F&F'F(F&*&-F-6#,&F'F&-F-6#F'F&F&F/F(F(,&*&F&F&F'F( F&*&-F-6#,&F'F&-F-6#F'F&F&F/F(F&,&*&F&F&F'F(F&*&-F-6#,&F'F&-F-6#F'F(F& F/F(F(,&*&F&F&F'F(F&*&-F-6#,&F'F&-F-6#F'F(F&F/F(F&" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 39 "The maximum points all lie on the line \+ " }{XPPEDIT 18 0 "y=1" "6#/%\"yG\"\"\"" }{TEXT -1 39 " and the minimum points all lie on the " }{TEXT 377 1 "x" }{TEXT -1 7 " axis. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "g := x -> 4*x*(1-x):\n'g(g(g(x)))'=g(g(g(x)));\n``=expand(rhs(%)); \n``=factor(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"gG6#-F%6# -F%6#%\"xG,$*,\"#k\"\"\"F+F/,&F/F/F+!\"\"F/,&F/F/*(\"\"%F/F+F/F0F/F1F/ ,&F/F/**\"#;F/F+F/F0F/F2F/F1F/F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% !G,2*&\"#k\"\"\"%\"xGF(F(*&\"%W8F()F)\"\"#F(!\"\"*&\"&_2\"F()F)\"\"$F( F(*&\"&SA%F()F)\"\"%F(F.*&\"&7,*F()F)\"\"&F(F(*&\"''\\1\"F()F)\"\"'F(F .*&\"&Ob'F()F)\"\"(F(F(*&\"&%Q;F()F)\"\")F(F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$*,\"#k\"\"\"%\"xGF(,&F(!\"\"F)F(F(),&*&\"\"#F(F)F (F(F(F+F/F(),(*&\"\")F()F)F/F(F(*&F3F(F)F(F+F(F(F/F(F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "Diff('g (g(g(x)))',x)=Diff(rhs(%),x);\n``=value(rhs(%));\n``=expand(rhs(%));\n ``=factor(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"g G6#-F(6#-F(6#%\"xGF.-F%6$,$*,\"#k\"\"\"F.F4,&F.F4F4!\"\"F4),&*&\"\"#F4 F.F4F4F4F6F:F4),(*&\"\")F4)F.F:F4F4*&F>F4F.F4F6F4F4F:F4F6F." }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/%!G,***\"#k\"\"\",&%\"xGF(F(!\"\"F(),&*&\" \"#F(F*F(F(F(F+F/F(),(*&\"\")F()F*F/F(F(*&F3F(F*F(F+F(F(F/F(F+**F'F(F* F(F,F(F0F(F+*,\"$c#F(F*F(F)F(F-F(F0F(F+*.\"$G\"F(F*F(F)F(F,F(F1F(,&*& \"#;F(F*F(F(F3F+F(F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,2*&\"'g0X \"\"\")%\"xG\"\"%F(F(*&\"'w*Q'F()F*\"\"&F(!\"\"*&\"'_(e%F()F*\"\"'F(F( *&\"'s58F()F*\"\"(F(F0*&\"'g*o\"F()F*\"\"$F(F0*&\"&cA$F()F*\"\"#F(F(*& \"%)o#F(F*F(F0\"#kF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,$**\"#k\" \"\",&*&\"\"#F(%\"xGF(F(F(!\"\"F(,(*&\"\")F()F,F+F(F(*&F0F(F,F(F-F(F(F (,,*&\"$G\"F()F,\"\"%F(F(*&\"$c#F()F,\"\"$F(F-*&\"$g\"F(F1F(F(*&\"#KF( F,F(F-F(F(F(F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 68 "g := x -> 4*x*(1-x):\nsolve(diff(g(g(g(x))),x) );\nevalf(evalf[14](%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)#\"\"\"\" \"#,&F#F$*&\"\"%!\"\"F%F#F$,&F#F$*&F(F)F%F#F),&F#F$*&F(F),&F%F$*$F%F#F $F#F),&F#F$*&F(F)F.F#F$,&F#F$*&F(F),&F%F$F/F)F#F),&F#F$*&F(F)F4F#F$" } }{PARA 11 "" 1 "" {XPPMATH 20 "6)$\"+++++]!#5$\"+1R`N&)F%$\"+%4mWY\"F% $\"+uL-1Q!#6$\"+jwR>'*F%$\"+QGe'3$F%$\"+irT8pF%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 "The graph of " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 51 " has four maximum \+ points and three minimum points. 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So f or example:" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d/dx; " "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[x^2+x^3 ];" "6#7#,&*$%\"xG\"\"#\"\"\"*$F&\"\"$F(" }{TEXT -1 4 " = " } {XPPEDIT 18 0 "2*x+3*x^2;" "6#,&*&\"\"#\"\"\"%\"xGF&F&*&\"\"$F&*$F'F%F &F&" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 53 "However, we can ea sily check that differentiation is " }{TEXT 259 39 "not interchangeabl e with multiplication" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 12 "For example," }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d/d x;" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[x^2*` .`*x^3] = d/dx;" "6#/7#*(%\"xG\"\"#%\".G\"\"\"F&\"\"$*&%\"dGF)%#dxG!\" \"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[x^5];" "6#7#*$%\"xG\"\"&" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "5*x^4;" "6#*&\"\"&\"\"\"*$%\"xG\"\"%F%" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 24 "while, on the other han d" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG \"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[x^2];" "6#7#*$%\"xG \"\"#" }{TEXT -1 1 " " }{TEXT 323 1 "." }{TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx;" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 " [x^3];" "6#7#*$%\"xG\"\"$" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "2*x;" "6# *&\"\"#\"\"\"%\"xGF%" }{TEXT -1 1 " " }{TEXT 322 1 "." }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "3*x^2;" "6#*&\"\"$\"\"\"*$%\"xG\"\"#F%" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "6*x^3;" "6#*&\"\"'\"\"\"*$%\"xG\"\"$F%" }{TEXT -1 4 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "Instead the following product rule applies." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "1. " }{TEXT 272 37 "The pr oduct rule in function notation" }{TEXT 271 2 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 6 "Given " }{XPPEDIT 18 0 "f (x) = u(x)*v(x)" "6#/-%\"fG6#%\"xG*&-%\"uG6#F'\"\"\"-%\"vG6#F'F," } {TEXT -1 8 ", where " }{XPPEDIT 18 0 "u(x)" "6#-%\"uG6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v(x)" "6#-%\"vG6#%\"xG" }{TEXT -1 18 " ha ve derivatives " }{XPPEDIT 18 0 "`u '`(x)" "6#-%$u~'G6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "`v '`(x)" "6#-%$v~'G6#%\"xG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "`f '`(x)=`u '`(x)*v(x)+u(x)*`v '`(x)" "6#/-%$f~'G6#% \"xG,&*&-%$u~'G6#F'\"\"\"-%\"vG6#F'F-F-*&-%\"uG6#F'F--%$v~'G6#F'F-F-" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 277 17 "___ ______________" }{TEXT -1 1 " " }{TEXT 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "In words:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 273 79 "To differentiate a product of two functions, form the su m of 2 terms as follows" }{TEXT -1 3 ":\n\n" }{TEXT 310 8 "1st term" } {TEXT -1 4 ": " }{TEXT 259 57 "derivative of 1st function times 2nd \+ function (unchanged)" }{TEXT -1 2 "\n\n" }{TEXT 311 8 "2nd term" } {TEXT -1 3 ": " }{TEXT 259 57 "1st function (unchanged) times derivat ive of 2nd function" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 28 "Thus, for the example above " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" } {TEXT -1 1 " " }{XPPEDIT 18 0 "[x^2*`.`*x^3] = d/dx;" "6#/7#*(%\"xG\" \"#%\".G\"\"\"F&\"\"$*&%\"dGF)%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[x^2]*`.`*x^3+x^2" "6#,&*(7#*$%\"xG\"\"#\"\"\"%\".GF)F'\"\"$F)*$ F'F(F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\" \"" }{TEXT -1 2 " [" }{XPPEDIT 18 0 "x^3" "6#*$%\"xG\"\"$" }{TEXT -1 4 "] = " }{XPPEDIT 18 0 "[2*x]*`.`*x^3+x^2*`.`*[3*x^2] = 5*x^4;" "6#/, &*(7#*&\"\"#\"\"\"%\"xGF)F)%\".GF)F*\"\"$F)*(F*F(F+F)7#*&F,F)*$F*F(F)F )F)*&\"\"&F)*$F*\"\"%F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 12 "as expected." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "2. " }{TEXT 275 47 "The product rule in the Leibniz or \"d \" notation" }{TEXT 274 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 6 "Given " }{XPPEDIT 18 0 "y = u*v" "6#/%\"yG *&%\"uG\"\"\"%\"vGF'" }{TEXT -1 8 ", where " }{TEXT 325 1 "u" }{TEXT -1 5 " and " }{TEXT 326 1 "v" }{TEXT -1 33 " are differentiable functi ons of " }{TEXT 324 1 "x" }{TEXT -1 1 "," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = u" "6#/*&%#dyG\"\"\"%#dxG!\"\"%\"uG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dv/dx+v;" "6#,&*&%#dvG\"\"\"%#dxG!\"\" F&%\"vGF&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "du/dx" "6#*&%#duG\"\"\"%#dx G!\"\"" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 278 10 "__________" }{TEXT -1 1 " " }{TEXT 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "Note that the terms are in a di fferent order from that given in the function form for the rule." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 258 31 "Explanation of the product rule" }{TEXT 276 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 " Given a product " }{XPPEDIT 18 0 "y = u(x)*v(x)" "6#/%\"yG*&-%\"uG6#% \"xG\"\"\"-%\"vG6#F)F*" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "u=u(x) " "6#/%\"uG-F$6#%\"xG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v=v(x)" "6# /%\"vG-F$6#%\"xG" }{TEXT -1 33 " are differentiable functions of " } {TEXT 340 1 "x" }{TEXT -1 7 ", let " }{XPPEDIT 18 0 "Delta;" "6#%&Del taG" }{TEXT 341 1 "x" }{TEXT -1 30 " be a change in the variable " } {TEXT 342 1 "x" }{TEXT -1 9 " and let " }{XPPEDIT 18 0 "Delta;" "6#%&D eltaG" }{TEXT 343 1 "u" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "Delta;" "6#%& DeltaG" }{TEXT 344 1 "v" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "Delta;" " 6#%&DeltaG" }{TEXT 345 1 "y" }{TEXT -1 47 " be the corresponding chang es in the variables " }{XPPEDIT 18 0 "u ,v" "6$%\"uG%\"vG" }{TEXT -1 5 " and " }{TEXT 346 1 "y" }{TEXT -1 14 " respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "We use the same lett ers " }{TEXT 347 1 "u" }{TEXT -1 5 " and " }{TEXT 348 1 "v" }{TEXT -1 56 " both for the name, and the value, of the two functions." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "Thus, if we wri te " }{XPPEDIT 18 0 "f(x)" "6#-%\"fG6#%\"xG" }{TEXT -1 5 " for " } {TEXT 349 1 "y" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "f(x) = u(x)* v(x)" "6#/-%\"fG6#%\"xG*&-%\"uG6#F'\"\"\"-%\"vG6#F'F," }{TEXT -1 6 ", \+ then" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Delta;" "6#% &DeltaG" }{XPPEDIT 18 0 "u = u(x+h)-u(x);" "6#/%\"uG,&-F$6#,&%\"xG\"\" \"%\"hGF*F*-F$6#F)!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "Delta;" "6# %&DeltaG" }{XPPEDIT 18 0 "x = v(x+h)-v(x)" "6#/%\"xG,&-%\"vG6#,&F$\"\" \"%\"hGF*F*-F'6#F$!\"\"" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "Delta; " "6#%&DeltaG" }{XPPEDIT 18 0 "y = f(x+h)-f(x);" "6#/%\"yG,&-%\"fG6#,& %\"xG\"\"\"%\"hGF+F+-F'6#F*!\"\"" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 7 "where " }{XPPEDIT 18 0 "h = Delta;" "6#/%\"hG%&DeltaG" } {TEXT 350 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 7 "We have" }}{PARA 256 "" 0 "" {TEXT -1 2 " \+ " }{XPPEDIT 18 0 "y+Delta*y;" "6#,&%\"yG\"\"\"*&%&DeltaGF%F$F%F%" } {TEXT -1 5 " = " }{XPPEDIT 18 0 "(u+Delta*u)*(v+Delta*v);" "6#*&,&% \"uG\"\"\"*&%&DeltaGF&F%F&F&F&,&%\"vGF&*&F(F&F*F&F&F&" }{TEXT -1 1 " \+ " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "y+Delta*y;" "6#,&%\"yG\"\"\"*&%&D eltaGF%F$F%F%" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "u*v+u*Delta*v+v*Delta *u+Delta*u*Delta*v;" "6#,**&%\"uG\"\"\"%\"vGF&F&*(F%F&%&DeltaGF&F'F&F& *(F'F&F)F&F%F&F&**F)F&F%F&F)F&F'F&F&" }}{PARA 0 "" 0 "" {TEXT -1 9 "Th erefore" }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Delta*y; " "6#*&%&DeltaG\"\"\"%\"yGF%" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "u*Delt a*v+v*Delta*u+Delta*u*Delta*v;" "6#,(*(%\"uG\"\"\"%&DeltaGF&%\"vGF&F&* (F(F&F'F&F%F&F&**F'F&F%F&F'F&F(F&F&" }{TEXT -1 2 " " }}{PARA 0 "" 0 " " {TEXT -1 2 "so" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "D elta*y/(Delta*x);" "6#*(%&DeltaG\"\"\"%\"yGF%*&F$F%%\"xGF%!\"\"" } {TEXT -1 4 " = " }{TEXT 351 1 "u" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Del ta*v/(Delta*x);" "6#*(%&DeltaG\"\"\"%\"vGF%*&F$F%%\"xGF%!\"\"" }{TEXT -1 4 " + " }{TEXT 352 1 "v" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Delta*u/( Delta*x);" "6#*(%&DeltaG\"\"\"%\"uGF%*&F$F%%\"xGF%!\"\"" }{TEXT -1 4 " + " }{XPPEDIT 18 0 "Delta*u/(Delta*x);" "6#*(%&DeltaG\"\"\"%\"uGF%*& F$F%%\"xGF%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Delta*v/(Delta*x);" "6#*(%&DeltaG\"\"\"%\"vGF%*&F$F%%\"xGF%!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Delta*x" "6#*&%&DeltaG\"\"\"%\"xGF%" }{TEXT -1 2 " " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 3 "As " } {XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT 353 1 "x" }{TEXT -1 17 " a pproaches 0, " }{XPPEDIT 18 0 "Delta*u/(Delta*x);" "6#*(%&DeltaG\"\" \"%\"uGF%*&F$F%%\"xGF%!\"\"" }{TEXT -1 14 " approaches " }{XPPEDIT 18 0 "du/dx;" "6#*&%#duG\"\"\"%#dxG!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "Delta*v/(Delta*x);" "6#*(%&DeltaG\"\"\"%\"vGF%*&F$F%%\"xGF%!\"\" " }{TEXT -1 14 " approaches " }{XPPEDIT 18 0 "dv/dx;" "6#*&%#dvG\"\" \"%#dxG!\"\"" }{TEXT -1 13 ". Note that " }{XPPEDIT 18 0 "Delta*u/(De lta*x)" "6#*(%&DeltaG\"\"\"%\"uGF%*&F$F%%\"xGF%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Delta*v/(Delta*x)" "6#*(%&DeltaG\"\"\"%\"vGF%*&F$F%%\" xGF%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "Delta*x" "6#*&%&DeltaG\"\" \"%\"xGF%" }{TEXT -1 15 " approaches 0. " }}{PARA 0 "" 0 "" {TEXT -1 66 "It follows that the right hand side of the last equation tends to \+ " }{TEXT 354 1 "u" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dv/dx+v;" "6#,&*&%# dvG\"\"\"%#dxG!\"\"F&%\"vGF&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "du/dx;" "6#*&%#duG\"\"\"%#dxG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "Therefore the left hand side al so tends to this value, so we conclude that" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = u;" "6#/*&%#dyG\"\"\"%#dxG!\"\" %\"uG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dv/dx+v;" "6#,&*&%#dvG\"\"\"%#d xG!\"\"F&%\"vGF&" }{TEXT -1 1 " " }{XPPEDIT 18 0 "du/dx;" "6#*&%#duG\" \"\"%#dxG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "Product rule examples" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 100 "Maple shows how \+ to differentiate a product, if we ensure that the functions f and g ar e unassigned.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "f := 'f': g := 'g': x := 'x':\nDiff(f(x)*g(x),x);\nvalue(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%%DiffG6$*&-%\"fG6#%\"xG\"\"\"-%\"gGF)F+F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&-%%diffG6$-%\"fG6#%\"xGF+\"\"\"-%\"gGF*F ,F,*&F(F,-F&6$F-F+F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 7 "Given " }{XPPEDIT 18 0 "f(x) = x^2-3*x+4;" "6#/-%\"fG6#%\"xG,(*$F'\"\"#\"\"\"*&\"\"$F+F'F+! \"\"\"\"%F+" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "g(x) = x^2-1;" "6#/- %\"gG6#%\"xG,&*$F'\"\"#\"\"\"F+!\"\"" }{TEXT -1 52 ", we can perform t he differentiation of the product " }{XPPEDIT 18 0 "f(x)*g(x);" "6#*&- %\"fG6#%\"xG\"\"\"-%\"gG6#F'F(" }{TEXT -1 41 " in two ways.\n\nWe can \+ expand the product " }{XPPEDIT 18 0 "f(x)*g(x);" "6#*&-%\"fG6#%\"xG\" \"\"-%\"gG6#F'F(" }{TEXT -1 30 " and then differentiate . . .\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "f := x ->x^2-3*x+4:\ng := x- >x^2-1:\nf(x)*g(x);\nexpand(%);\nDiff(%,x);\nvalue(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,(*$)%\"xG\"\"#\"\"\"F)*&\"\"$F)F'F)!\"\"\"\"%F) F),&F%F)F)F,F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*$)%\"xG\"\"%\"\" \"F(*&\"\"$F()F&\"\"#F(F(*&F*F()F&F*F(!\"\"*&F*F(F&F(F(F'F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$,,*$)%\"xG\"\"%\"\"\"F+*&\"\"$F+)F )\"\"#F+F+*&F-F+)F)F-F+!\"\"*&F-F+F)F+F+F*F2F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&\"\"%\"\"\")%\"xG\"\"$F&F&*&\"\"'F&F(F&F&*&\"\"*F&) F(\"\"#F&!\"\"F)F&" }}}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x^4-3*x^3+3*x^2+3*x-4" "6#,,*$%\"xG\"\"%\"\"\"*&\"\"$F'*$F%F)F'!\"\"* &F)F'*$F%\"\"#F'F'*&F)F'F%F'F'F&F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 55 " . . . or we can use the product rule to differentiate " }{XPPEDIT 18 0 "f(x)*g(x);" "6#*&-%\"fG6#%\"xG\"\"\"-%\"gG6#F'F(" } {TEXT -1 85 " and then, if we want to compare with the previous result , we can expand afterwards.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "Diff(f(x)*g(x),x);\nvalue(%);\nexpand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$*&,(*$)%\"xG\"\"#\"\"\"F,*&\"\"$F,F*F,!\"\"\" \"%F,F,,&F(F,F,F/F,F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&*&\"\"# \"\"\"%\"xGF(F(\"\"$!\"\"F(,&*$)F)F'F(F(F(F+F(F(*(F'F(,(F-F(*&F*F(F)F( F+\"\"%F(F(F)F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**&\"\"%\"\"\")% \"xG\"\"$F&F&*&\"\"'F&F(F&F&*&\"\"*F&)F(\"\"#F&!\"\"F)F&" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 0 "" {TEXT 258 8 "Question" } {TEXT 358 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 7 "Given " }{XPPEDIT 18 0 "f(x) = (x+1/x)*(sqrt(x)-2);" "6#/-%\"fG6#%\"xG*&,&F'\"\"\"*&F*F*F'! \"\"F*F*,&-%%sqrtG6#F'F*\"\"#F,F*" }{TEXT -1 7 ", find " }{XPPEDIT 18 0 "`f '`(x)" "6#-%$f~'G6#%\"xG" }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT 269 8 "Solution" }{TEXT 359 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "u(x) = x+1/x;" "6#/-%\"uG6#%\"xG,&F'\"\"\"*&F)F )F'!\"\"F)" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "v(x) = sqrt(x)-2;" " 6#/-%\"vG6#%\"xG,&-%%sqrtG6#F'\"\"\"\"\"#!\"\"" }{TEXT -1 10 ", so tha t " }{XPPEDIT 18 0 "f(x) = u(x)*v(x);" "6#/-%\"fG6#%\"xG*&-%\"uG6#F'\" \"\"-%\"vG6#F'F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 8 "We hav e " }{XPPEDIT 18 0 "`u '`(x) = 1-1/(x^2);" "6#/-%$u~'G6#%\"xG,&\"\"\"F )*&F)F)*$F'\"\"#!\"\"F-" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "`v '`(x) = 1/(2*sqrt(x));" "6#/-%$v~'G6#%\"xG*&\"\"\"F)*&\"\"#F)-%%sqrtG6#F'F) !\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 27 "Then, by the pr oduct rule, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '` (x)=`u '`(x)*v(x)+u(x)*`v '`(x)" "6#/-%$f~'G6#%\"xG,&*&-%$u~'G6#F'\"\" \"-%\"vG6#F'F-F-*&-%\"uG6#F'F--%$v~'G6#F'F-F-" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "``=(1-1/(x^2))*`.`*(sqrt(x)-2)+(x+1/x)*`.`*``(1/(2*sqrt (x)))" "6#/%!G,&*(,&\"\"\"F(*&F(F(*$%\"xG\"\"#!\"\"F-F(%\".GF(,&-%%sqr tG6#F+F(F,F-F(F(*(,&F+F(*&F(F(F+F-F(F(F.F(-F$6#*&F(F(*&F,F(-F16#F+F(F- F(F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "``=(1-x^(-2))*`.`*(x^(1/2)-2)+1/2" "6#/%!G,&*(,&\" \"\"F()%\"xG,$\"\"#!\"\"F-F(%\".GF(,&)F**&F(F(F,F-F(F,F-F(F(*&F(F(F,F- F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "(x+x^(-1))*`.`*``(x^(-1/2))" "6#*( ,&%\"xG\"\"\")F%,$F&!\"\"F&F&%\".GF&-%!G6#)F%,$*&F&F&\"\"#F)F)F&" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=x^(1/2)-2-x^(-3/2)+2*x^(-2)+1/2" "6# /%!G,,)%\"xG*&\"\"\"F)\"\"#!\"\"F)F*F+)F',$*&\"\"$F)F*F+F+F+*&F*F))F', $F*F+F)F)*&F)F)F*F+F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^(1/2)+1/2" "6 #,&)%\"xG*&\"\"\"F'\"\"#!\"\"F'*&F'F'F(F)F'" }{TEXT -1 1 " " } {XPPEDIT 18 0 "x^(-3/2)" "6#)%\"xG,$*&\"\"$\"\"\"\"\"#!\"\"F*" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=3/2" " 6#/%!G*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^(1/2) - 2 -1/2" "6#,()%\"xG*&\"\"\"F'\"\"#!\"\"F'F(F)*&F'F'F(F)F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^(-3/2) + 2*x^(-2)" "6#,&)%\"xG,$*&\"\"$\"\" \"\"\"#!\"\"F+F)*&F*F))F%,$F*F+F)F)" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``= 3/2" "6#/%!G*&\"\"$\"\"\"\"\"#! \"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(x)-2-1/(2*x*sqrt(x))+2/x^2 " "6#,*-%%sqrtG6#%\"xG\"\"\"\"\"#!\"\"*&F(F(*(F)F(F'F(-F%6#F'F(F*F**&F )F(*$F'F)F*F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 15 "Alternatively, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "f(x)=(x+1/x)*(sqrt(x)-2)" "6#/-%\"fG6#% \"xG*&,&F'\"\"\"*&F*F*F'!\"\"F*F*,&-%%sqrtG6#F'F*\"\"#F,F*" }{XPPEDIT 18 0 "``=(x+x^(-1))*(x^(1/2)-2)" "6#/%!G*&,&%\"xG\"\"\")F',$F(!\"\"F(F (,&)F'*&F(F(\"\"#F+F(F/F+F(" }{XPPEDIT 18 0 "``=x^(3/2)-2*x+x^(-1/2)-2 *x^(-1)" "6#/%!G,*)%\"xG*&\"\"$\"\"\"\"\"#!\"\"F**&F+F*F'F*F,)F',$*&F* F*F+F,F,F**&F+F*)F',$F*F,F*F," }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "so " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x) = 3/2;" "6#/-%$f~'G6#%\"xG*&\"\"$\"\"\"\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^(1/2)-2-1/2" "6#,()%\"xG*&\"\"\"F'\"\"#!\"\"F'F( F)*&F'F'F(F)F)" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x^(-3/2)+2*x^(-2)" "6# ,&)%\"xG,$*&\"\"$\"\"\"\"\"#!\"\"F+F)*&F*F))F%,$F*F+F)F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "Diff((x+1/x)*(sqrt(x)-2),x);\nvalue(%);\nexpand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$*&,&%\"xG\"\"\"*&F)F)F(!\" \"F)F),&*$F(#F)\"\"#F)F/F+F)F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*& ,&\"\"\"F&*&F&F&*$)%\"xG\"\"#F&!\"\"F,F&,&*$F*#F&F+F&F+F,F&F&*(F+F,,&F *F&*&F&F&F*F,F&F&F*#F,F+F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**(\"\" $\"\"\"\"\"#!\"\"%\"xG#F&F'F&F'F(*&F&F&*&F'F&)F)#F%F'F&F(F(*&F'F&F)!\" #F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 0 "" {TEXT 258 8 "Question" }{TEXT 297 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 7 "Given " }{XPPEDIT 18 0 "f(x) = (x^4+7)*sqrt(x^3-4*x+3);" "6#/-%\"fG6 #%\"xG*&,&*$F'\"\"%\"\"\"\"\"(F,F,-%%sqrtG6#,(*$F'\"\"$F,*&F+F,F'F,!\" \"F3F,F," }{TEXT -1 7 ", find " }{XPPEDIT 18 0 "`f '`(x)" "6#-%$f~'G6# %\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 " " 0 "" {TEXT 269 8 "Solution" }{TEXT 298 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "u(x) = x^4+7;" "6#/-%\"uG6#%\"xG,&*$ F'\"\"%\"\"\"\"\"(F+" }{TEXT -1 7 " and " }{XPPEDIT 18 0 "v(x) = sqr t(x^3-4*x+3);" "6#/-%\"vG6#%\"xG-%%sqrtG6#,(*$F'\"\"$\"\"\"*&\"\"%F.F' F.!\"\"F-F." }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "f(x) = u(x)*v(x );" "6#/-%\"fG6#%\"xG*&-%\"uG6#F'\"\"\"-%\"vG6#F'F," }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 8 "We have " }{XPPEDIT 18 0 "`u '`(x) = 4*x^ 3;" "6#/-%$u~'G6#%\"xG*&\"\"%\"\"\"*$F'\"\"$F*" }{TEXT -1 6 " and " } {XPPEDIT 18 0 "`v '`(x) = (3*x^2-4)/(2*sqrt(x^3-4*x+3));" "6#/-%$v~'G6 #%\"xG*&,&*&\"\"$\"\"\"*$F'\"\"#F,F,\"\"%!\"\"F,*&F.F,-%%sqrtG6#,(*$F' F+F,*&F/F,F'F,F0F+F,F,F0" }{TEXT -1 20 ", by the chain rule." }}{PARA 0 "" 0 "" {TEXT -1 27 "Then, by the product rule, " }}{PARA 256 "" 0 " " {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x) = `u '`(x)*v(x)+u(x)*`v '`(x )" "6#/-%$f~'G6#%\"xG,&*&-%$u~'G6#F'\"\"\"-%\"vG6#F'F-F-*&-%\"uG6#F'F- -%$v~'G6#F'F-F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 4*x^3*`.`*sqrt(x ^3-4*x+3)+(x^4+7)*`.`*``((3*x^2-4)/(2*sqrt(x^3-4*x+3)));" "6#/%!G,&** \"\"%\"\"\"*$%\"xG\"\"$F(%\".GF(-%%sqrtG6#,(*$F*F+F(*&F'F(F*F(!\"\"F+F (F(F(*(,&*$F*F'F(\"\"(F(F(F,F(-F$6#*&,&*&F+F(*$F*\"\"#F(F(F'F3F(*&F>F( -F.6#,(*$F*F+F(*&F'F(F*F(F3F+F(F(F3F(F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " `` = (8*x^3*(x^3-4*x+3)+(x^4+7)*(3*x^2-4))/(2*sqrt(x^3-4*x+3));" "6#/% !G*&,&*(\"\")\"\"\"*$%\"xG\"\"$F),(*$F+F,F)*&\"\"%F)F+F)!\"\"F,F)F)F)* &,&*$F+F0F)\"\"(F)F),&*&F,F)*$F+\"\"#F)F)F0F1F)F)F)*&F9F)-%%sqrtG6#,(* $F+F,F)*&F0F)F+F)F1F,F)F)F1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "``=(8*x^6-3 2*x^4+24*x^3+3*x^6-4*x^4+21*x^2-28)/(2*sqrt(x^3-4*x+3))" "6#/%!G*&,0*& \"\")\"\"\"*$%\"xG\"\"'F)F)*&\"#KF)*$F+\"\"%F)!\"\"*&\"#CF)*$F+\"\"$F) F)*&F5F)*$F+F,F)F)*&F0F)*$F+F0F)F1*&\"#@F)*$F+\"\"#F)F)\"#GF1F)*&F=F)- %%sqrtG6#,(*$F+F5F)*&F0F)F+F)F1F5F)F)F1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " ``=(11*x^6-36*x^4+24*x^3+21*x^2-28)/(2*sqrt(x^3-4*x+3))" "6#/%!G*&,,*& \"#6\"\"\"*$%\"xG\"\"'F)F)*&\"#OF)*$F+\"\"%F)!\"\"*&\"#CF)*$F+\"\"$F)F )*&\"#@F)*$F+\"\"#F)F)\"#GF1F)*&F9F)-%%sqrtG6#,(*$F+F5F)*&F0F)F+F)F1F5 F)F)F1" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "Diff((x^4+7)*sqrt(x^3-4*x+3),x);\nv alue(%);\nnormal(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$*&,& *$)%\"xG\"\"%\"\"\"F,\"\"(F,F,,(*$)F*\"\"$F,F,*&F+F,F*F,!\"\"F1F,#F,\" \"#F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*(\"\"%\"\"\")%\"xG\"\"$F&, (*$F'F&F&*&F%F&F(F&!\"\"F)F&#F&\"\"#F&**F/F-,&*$)F(F%F&F&\"\"(F&F&F*#F -F/,&*&F)F&)F(F/F&F&F%F-F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*(\" \"#!\"\",,*&\"#6\"\"\")%\"xG\"\"'F*F**&\"#OF*)F,\"\"%F*F&*&\"#CF*)F,\" \"$F*F**&\"#@F*)F,F%F*F*\"#GF&F*,(*$F4F*F**&F1F*F,F*F&F5F*#F&F%F*" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 0 "" {TEXT 258 8 "Question" }{TEXT 299 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 7 "Given " } {XPPEDIT 18 0 "f(x) = x*sqrt(1-x^2);" "6#/-%\"fG6#%\"xG*&F'\"\"\"-%%sq rtG6#,&F)F)*$F'\"\"#!\"\"F)" }{TEXT -1 7 ", find " }{XPPEDIT 18 0 "`f \+ '`(x)" "6#-%$f~'G6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "Sketch the graph " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG " }{TEXT -1 75 ", and find the coordinates of the maximum and minimum \+ points on the graph. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 269 8 "Solution" }{TEXT 300 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 20 "By the product rule," }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "`f '`(x) = d/dx;" "6#/-%$f~'G6#%\"xG*&%\"dG\"\"\"%#dxG! \"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[x]*`.`*``(sqrt(1-x^2))+x*`.`;" "6#,&*(7#%\"xG\"\"\"%\".GF'-%!G6#-%%sqrtG6#,&F'F'*$F&\"\"#!\"\"F'F'*&F &F'F(F'F'" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG !\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "[sqrt(1-x^2)]" "6#7#-%%sqrtG6#, &\"\"\"F(*$%\"xG\"\"#!\"\"" }{TEXT -1 5 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = 1*`.`*sqrt(1-x^2)+x*`.`*``(-x/sqrt (1-x^2));" "6#/%!G,&*(\"\"\"F'%\".GF'-%%sqrtG6#,&F'F'*$%\"xG\"\"#!\"\" F'F'*(F.F'F(F'-F$6#,$*&F.F'-F*6#,&F'F'*$F.F/F0F0F0F'F'" }{TEXT -1 1 " \+ " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "`` = sqrt(1-x^2) -x^2/sqrt(1-x^2);" "6#/%!G,&-%%sqrtG6#,&\"\"\"F**$%\"xG\"\"#!\"\"F**&F ,F--F'6#,&F*F**$F,F-F.F.F." }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (1-x^2-x^2)/sqrt(1-x^2);" "6#/%!G*&,(\" \"\"F'*$%\"xG\"\"#!\"\"*$F)F*F+F'-%%sqrtG6#,&F'F'*$F)F*F+F+" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`` = (1-2* x^2)/sqrt(1-x^2);" "6#/%!G*&,&\"\"\"F'*&\"\"#F'*$%\"xGF)F'!\"\"F'-%%sq rtG6#,&F'F'*$F+F)F,F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "f := x -> x*sqrt(1-x^2);\nDiff(f(x),x);\nvalue(%);\nnormal(%); \ndf := unapply(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\" xG6\"6$%)operatorG%&arrowGF(*&9$\"\"\"-%%sqrtG6#,&F.F.*$)F-\"\"#F.!\" \"F.F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$*&%\"xG\"\"\"- %%sqrtG6#,&F(F(*$)F'\"\"#F(!\"\"F(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,&*$-%%sqrtG6#,&\"\"\"F)*$)%\"xG\"\"#F)!\"\"F)F)*&*$F+F)F)*$-F&6#F(F )F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&!\"\"\"\"\"*$)%\"xG\"\" #F'F+F'*$-%%sqrtG6#,&F'F'F(F&F'F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#dfGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&,&!\"\"\"\"\"*$)9$\"\"#F 0F4F0*$-%%sqrtG6#,&F0F0F1F/F0F/F/F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "plot(x*sqrt(1-x^2),x= -1..1,y);" }}{PARA 13 "" 1 "" {GLPLOT2D 354 201 201 {PLOTDATA 2 "6%-%' CURVESG6$7co7$$!\"\"\"\"!F*7$$!1n;HdNvs**!#;$!1$))G)H-!oN(!#<7$$!1MLe9 r]X**F.$!1'zTCAeo.\"F.7$$!1,](=ng#=**F.$!1,>kiDal7F.7$$!1nm;HU,\"*)*F. $!1M#fT?6jX\"F.7$$!1++vV8_O)*F.$!1Rv=4=Nr9zlOE$F.7$$!1LL$3\"\\F.7$$!1nm;/R=0vF.$!1QKW2vzf\\F.7 $$!1LL$3i_+I(F.$!1=So\"4f\"*)\\F.7$$!1++]P8#\\4(F.$!1=!\\j\"e))**\\F.7 $$!1ML$3FuF)oF.$!1\"4+2!>4$*\\F.7$$!1nm;/siqmF.$!1Hu/&oF'p\\F.7$$!1++] (y$pZiF.$!1&y=Xld#y[F.7$$!1LLL$yaE\"eF.$!1UX'oYQ)HZF.7$$!1nmm\">s%HaF. $!1^%pNh!\\fXF.7$$!1+++]$*4)*\\F.$!1#R$fI$H!HVF.7$$!1+++]_&\\c%F.$!1WM DU!e:1%F.7$$!1+++]1aZTF.$!18lw(Q')Rx$F.7$$!1nm;/#)[oPF.$!1C$o\"zbl!\\$ F.7$$!1MLL$=exJ$F.$!1'[IjeM)HJF.7$$!1MLLL2$f$HF.$!1/))38da1GF.7$$!1++] PYx\"\\#F.$!1!3izoyJT#F.7$$!1MLLL7i)4#F.$!1`P^;q)=0#F.7$$!1++]P'psm\"F .$!1Q)yk.LRk\"F.7$$!1++]74_c7F.$!1v5N!>ilC\"F.7$$!1JLL$3x%z#)F1$!1VwE$ []5D)F17$$!1MLL3s$QM%F1$!13n4(4P(RVF17$$!1^omm;zr)*!#>$!15J_coyr)*Fdv7 $$\"1$*))\\F.7$$\"1mm;HYt7vF.$\"1;['RgP$e\\F.7$$\"1LLek6,1xF.$ \"1Nr>Ex<6\\F.7$$\"1*******p(G**yF.$\"1fV3dA$Q%[F.7$$\"1mmmT6KU$)F.$\" 1/tF#)z1+YF.7$$\"1LLLLbdQ()F.$\"1-vl$[Q&[UF.7$$\"1++]i`1h\"*F.$\"1cz!e 0\"*Hn$F.7$$\"1++++PDj$*F.$\"1d7+KHv(G$F.7$$\"1++]P?Wl&*F.$\"1/Z@a>;*y #F.7$$\"1+]7G:3u'*F.$\"1j0^))=p\\CF.7$$\"1++v=5s#y*F.$\"1JT$pb.#G?F.7$ $\"1+D1k2/P)*F.$\"1V(z#e?lo " 0 "" {MPLTEXT 1 0 40 "xc := [so lve(df(x)=0)];\nyc := map(f,xc);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #xcG7$,$*$-%%sqrtG6#\"\"#\"\"\"#F,F+,$F'#!\"\"F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ycG7$#\"\"\"\"\"##!\"\"F(" }}}{PARA 0 "" 0 "" {TEXT -1 22 "The maximum point is " }{XPPEDIT 18 0 "[sqrt(2)/2, 1/2];" "6#7 $*&-%%sqrtG6#\"\"#\"\"\"F(!\"\"*&F)F)F(F*" }{TEXT -1 26 " and the mini mum point is " }{XPPEDIT 18 0 "[-sqrt(2)/2, -1/2];" "6#7$,$*&-%%sqrtG6 #\"\"#\"\"\"F)!\"\"F+,$*&F*F*F)F+F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 64 "The quotient rule for differentiation - explanation and examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 70 "Suppose that we need to find the derivative of a q uotient of the form " }{XPPEDIT 18 0 "f(x)=u(x)/v(x)" "6#/-%\"fG6#%\"x G*&-%\"uG6#F'\"\"\"-%\"vG6#F'!\"\"" }{TEXT -1 10 ", where u(" }{TEXT 330 1 "x" }{TEXT -1 8 ") and v(" }{TEXT 331 1 "x" }{TEXT -1 34 ") are \+ differentiable functions of " }{TEXT 355 1 "x" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 47 "Such a quotient can be expressed as a pro duct " }{XPPEDIT 18 0 "f(x) = u(x)*v(x)^(-1);" "6#/-%\"fG6#%\"xG*&-% \"uG6#F'\"\"\")-%\"vG6#F',$F,!\"\"F," }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "Then we can use the pro duct and chain rules to obtain an expression for the derivative " } {XPPEDIT 18 0 "`f '`(x)" "6#-%$f~'G6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "By the product \+ rule, we have" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f ' `(x) = `u '`(x)*v(x)^(-1)+u(x)" "6#/-%$f~'G6#%\"xG,&*&-%$u~'G6#F'\"\" \")-%\"vG6#F',$F-!\"\"F-F--%\"uG6#F'F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d/dx" "6#*&%\"dG\"\"\"%#dxG!\"\"" }{TEXT -1 2 " [" }{XPPEDIT 18 0 " v(x)^(-1)" "6#)-%\"vG6#%\"xG,$\"\"\"!\"\"" }{TEXT -1 2 "]." }}{PARA 0 "" 0 "" {TEXT -1 42 "The chain rule can then be used to obtain " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x)=`u '`(x)*v(x )^(-1)+u(x)*(-1)*v(x)^(-2)*`u '`(x)" "6#/-%$f~'G6#%\"xG,&*&-%$u~'G6#F' \"\"\")-%\"vG6#F',$F-!\"\"F-F-**-%\"uG6#F'F-,$F-F3F-)-F06#F',$\"\"#F3F --F+6#F'F-F-" }{TEXT -1 2 ", " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x) = `u '`(x)/v (x)-u(x)*`v '`(x)/(v(x)^2);" "6#/-%$f~'G6#%\"xG,&*&-%$u~'G6#F'\"\"\"-% \"vG6#F'!\"\"F-*(-%\"uG6#F'F--%$v~'G6#F'F-*$-F/6#F'\"\"#F1F1" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x) = (`u '`(x)*v(x)-u(x)*`v '`(x))/(v(x )^2);" "6#/-%$f~'G6#%\"xG*&,&*&-%$u~'G6#F'\"\"\"-%\"vG6#F'F.F.*&-%\"uG 6#F'F.-%$v~'G6#F'F.!\"\"F.*$-F06#F'\"\"#F9" }{TEXT -1 3 ". " }}{PARA 256 "" 0 "" {TEXT 301 18 "__________________" }{TEXT -1 1 " " }{TEXT 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 302 75 "To differentiate the quotient of two functions, form a quotient as f ollows" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" {TEXT 267 1 "\n" }{TEXT 269 23 "numerator of derivative" }{TEXT -1 1 ":" }{TEXT 256 2 " " }} {PARA 0 "" 0 "" {TEXT 256 6 " " }{TEXT 259 85 "[derivative of top times bottom (unchanged)] minus [top times derivative of bottom]" } {TEXT 256 1 "\n" }{TEXT 259 74 "______________________________________ ____________________________________" }{TEXT 256 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 269 25 "denominator of deriva tive" }{TEXT -1 1 ":" }{TEXT 256 3 " " }{TEXT 259 19 "denominator sq uared" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 71 " The quotient rule can also be expressed in the Leibniz or \"d\" notati on." }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "y = u/v" "6#/ %\"yG*&%\"uG\"\"\"%\"vG!\"\"" }{TEXT -1 8 ", where " }{TEXT 328 1 "u" }{TEXT -1 5 " and " }{TEXT 329 1 "v" }{TEXT -1 33 " are differentiable functions of " }{TEXT 327 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 4 "Then" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d y/dx = (v*``(du/dx)-u*``(dv/dx))/(v^2);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*& ,&*&%\"vGF&-%!G6#*&%#duGF&F'F(F&F&*&%\"uGF&-F.6#*&%#dvGF&F'F(F&F(F&*$F ,\"\"#F(" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT 303 12 "__________ __" }{TEXT -1 1 " " }{TEXT 0 0 "" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examp le 1" }}{PARA 257 "" 0 "" {TEXT 258 8 "Question" }{TEXT 304 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 7 " Given " }{XPPEDIT 18 0 "f(x) = (x^2-3)/(5 *x+1);" "6#/-%\"fG6#%\"xG*&,&*$F'\"\"#\"\"\"\"\"$!\"\"F,,&*&\"\"&F,F'F ,F,F,F,F." }{TEXT -1 7 ", find " }{XPPEDIT 18 0 "`f '`(x)" "6#-%$f~'G6 #%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 " " 0 "" {TEXT 269 8 "Solution" }{TEXT 305 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "u(x)=x^2-3" "6#/-%\"uG6#%\"xG,&*$F' \"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v(x)=5*x+1 " "6#/-%\"vG6#%\"xG,&*&\"\"&\"\"\"F'F+F+F+F+" }{TEXT -1 9 " so that " }{XPPEDIT 18 0 "`u '`(x) = 2*x;" "6#/-%$u~'G6#%\"xG*&\"\"#\"\"\"F'F*" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "`v '`(x) = 5;" "6#/-%$v~'G6#%\"xG \"\"&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x) = (`u '`(x)*v(x)-u (x)*`v '`(x))/(v(x)^2);" "6#/-%$f~'G6#%\"xG*&,&*&-%$u~'G6#F'\"\"\"-%\" vG6#F'F.F.*&-%\"uG6#F'F.-%$v~'G6#F'F.!\"\"F.*$-F06#F'\"\"#F9" }{TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 4 " = " }{XPPEDIT 18 0 "(2*x*` .`*(5*x+1)-(x^2-3)*`.`*5)/((5*x+1)^2);" "6#*&,&**\"\"#\"\"\"%\"xGF'%\" .GF',&*&\"\"&F'F(F'F'F'F'F'F'*(,&*$F(F&F'\"\"$!\"\"F'F)F'F,F'F1F'*$,&* &F,F'F(F'F'F'F'F&F1" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 4 " \+ = " }{XPPEDIT 18 0 "(10*x^2+2*x-5*x^2+15)/((5*x+1)^2);" "6#*&,**&\"#5 \"\"\"*$%\"xG\"\"#F'F'*&F*F'F)F'F'*&\"\"&F'*$F)F*F'!\"\"\"#:F'F'*$,&*& F-F'F)F'F'F'F'F*F/" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 4 " = " }{XPPEDIT 18 0 "(5*x^2+2*x+15)/((5*x+1)^2);" "6#*&,(*&\"\"&\"\"\"* $%\"xG\"\"#F'F'*&F*F'F)F'F'\"#:F'F'*$,&*&F&F'F)F'F'F'F'F*!\"\"" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Diff((x^2-3)/(5*x+1),x);\nvalue(%);\nnormal(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$*&,&*$)%\"xG\"\"#\"\"\"F ,!\"$F,F,,&F*\"\"&F,F,!\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&% \"xG\"\"\",&F%\"\"&F&F&!\"\"\"\"#*&,&*$)F%F*F&F&!\"$F&F&*$)F'F*F&F)!\" &" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,(*$)%\"xG\"\"#\"\"\"\"\"&F'F( \"#:F)F)*$),&F'F*F)F)F(F)!\"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 122 "Note that the initial expression obtaine d by Maple is the result of applying the product and chain rules to di fferentiate " }{XPPEDIT 18 0 "(x^2-3)*(5*x+1)^(-1)" "6#*&,&*$%\"xG\"\" #\"\"\"\"\"$!\"\"F(),&*&\"\"&F(F&F(F(F(F(,$F(F*F(" }{TEXT -1 1 "." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 257 " " 0 "" {TEXT 258 8 "Question" }{TEXT 306 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 7 " Given " }{XPPEDIT 18 0 "f(x) = sqrt(x)/(1-x^3);" "6#/-%\" fG6#%\"xG*&-%%sqrtG6#F'\"\"\",&F,F,*$F'\"\"$!\"\"F0" }{TEXT -1 7 ", fi nd " }{XPPEDIT 18 0 "`f '`(x)" "6#-%$f~'G6#%\"xG" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 269 8 "Solutio n" }{TEXT 307 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{XPPEDIT 18 0 "u(x) = sqrt(x);" "6#/-%\"uG6#%\"xG-%%sqrtG6#F'" }{TEXT -1 5 " and \+ " }{XPPEDIT 18 0 "v(x) = 1-x^3;" "6#/-%\"vG6#%\"xG,&\"\"\"F)*$F'\"\"$! \"\"" }{TEXT -1 9 " so that " }{XPPEDIT 18 0 "u*`'`(x) = 1/(2*sqrt(x)) ;" "6#/*&%\"uG\"\"\"-%\"'G6#%\"xGF&*&F&F&*&\"\"#F&-%%sqrtG6#F*F&!\"\" " }{TEXT -1 5 " and " }{XPPEDIT 18 0 "v*`'`(x) = -3*x^2;" "6#/*&%\"vG \"\"\"-%\"'G6#%\"xGF&,$*&\"\"$F&*$F*\"\"#F&!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x) = (`u '`(x)*v(x)-u(x)*`v '`(x))/(v(x)^2);" "6 #/-%$f~'G6#%\"xG*&,&*&-%$u~'G6#F'\"\"\"-%\"vG6#F'F.F.*&-%\"uG6#F'F.-%$ v~'G6#F'F.!\"\"F.*$-F06#F'\"\"#F9" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 4 " = " }{XPPEDIT 18 0 "(` `(1/(2*sqrt(x)))*`.`*(1-x^3)-sqrt(x)*`.`*(-3*x^2))/((1-x^3)^2);" "6#*& ,&*(-%!G6#*&\"\"\"F**&\"\"#F*-%%sqrtG6#%\"xGF*!\"\"F*%\".GF*,&F*F**$F0 \"\"$F1F*F**(-F.6#F0F*F2F*,$*&F5F**$F0F,F*F1F*F1F**$,&F*F**$F0F5F1F,F1 " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 4 " = " }{XPPEDIT 18 0 "(1-x^3+``(2*x)*``(3*x^2))/(2*sqrt(x) *(1-x^3)^2);" "6#*&,(\"\"\"F%*$%\"xG\"\"$!\"\"*&-%!G6#*&\"\"#F%F'F%F%- F,6#*&F(F%*$F'F/F%F%F%F%*(F/F%-%%sqrtG6#F'F%,&F%F%*$F'F(F)F/F)" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 4 " = " }{XPPEDIT 18 0 "(5*x^3+1)/(2*sqrt(x)*(1-x^3)^2);" "6 #*&,&*&\"\"&\"\"\"*$%\"xG\"\"$F'F'F'F'F'*(\"\"#F'-%%sqrtG6#F)F',&F'F'* $F)F*!\"\"F,F2" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "Diff(sqrt(x)/(1-x^3),x);\nva lue(%);\nnormal(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$*&*$- %%sqrtG6#%\"xG\"\"\"F,,&F,F,*$)F+\"\"$F,!\"\"F1F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"\"F%*&-%%sqrtG6#%\"xGF%,&F%F%*$)F*\"\"$F%!\"\"F %F/#F%\"\"#*&*$)F*#\"\"&F1F%F%*$)F+F1F%F/F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&\"\"\"F&*$)%\"xG\"\"$F&\"\"&F&*&-%%sqrtG6#F)F&),& !\"\"F&F'F&\"\"#F&F2#F&F3" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" } }}{PARA 257 "" 0 "" {TEXT 258 8 "Question" }{TEXT 360 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 7 "Given " }{XPPEDIT 18 0 "f(x) = x/(1+x^2);" "6#/ -%\"fG6#%\"xG*&F'\"\"\",&F)F)*$F'\"\"#F)!\"\"" }{TEXT -1 7 ", find " } {XPPEDIT 18 0 "`f '`(x)" "6#-%$f~'G6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "Sketch the grap h " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 75 ", an d find the coordinates of the maximum and minimum points on the graph. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 269 8 "So lution" }{TEXT 361 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 21 "By the quotie nt rule," }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`f '`(x) \+ = (``(1)*`.`*``(1+x^2)-``(x)*`.`*``(2*x))/((1+x^2)^2);" "6#/-%$f~'G6#% \"xG*&,&*(-%!G6#\"\"\"F.%\".GF.-F,6#,&F.F.*$F'\"\"#F.F.F.*(-F,6#F'F.F/ F.-F,6#*&F4F.F'F.F.!\"\"F.*$,&F.F.*$F'F4F.F4F;" }{TEXT -1 1 " " }} {PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "(1-x^2)/(1+x^2)^2" "6#*&,&\"\"\"F%*$%\"xG\"\"#!\"\"F%*$,&F%F%*$F'F(F%F(F)" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "f := x -> x/(1+x^2);\nDiff(f(x),x);\nvalue(%);\nnorma l(%);\nDf := unapply(%,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf* 6#%\"xG6\"6$%)operatorG%&arrowGF(*&9$\"\"\",&F.F.*$)F-\"\"#F.F.!\"\"F( F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$*&%\"xG\"\"\",&F(F(* $)F'\"\"#F(F(!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"\"F%,& F%F%*$)%\"xG\"\"#F%F%!\"\"F%*(F*F%F)F*F&!\"#F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&\"\"\"!\"\"*$)%\"xG\"\"#F&F&F&,&F&F&F(F&!\"#F'" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DfGf*6#%\"xG6\"6$%)operatorG%&arro wGF(,$*&,&\"\"\"!\"\"*$)9$\"\"#F/F/F/,&F/F/F1F/!\"#F0F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "pl ot(x/(1+x^2),x=-4..4,y);" }}{PARA 13 "" 1 "" {GLPLOT2D 354 201 201 {PLOTDATA 2 "6%-%'CURVESG6$7ao7$$!\"%\"\"!$!3?)eqkM)*HF-7$$!3!******\\`oz$GF1$!3Bf?V!yoW8 $F-7$$!3!omm;)3DoEF1$!3O&=?K?+iG$F-7$$!3?+++:v2*\\#F1$!3c3NiJu>\\MF-7$ $!3BLLL8>1DBF1$!3&fWF1$!3(f)*HWS74+%F-7$$!30++++@)f#=F1$!3bo1a@>&H@%F-7$$!3-+++gi, f;F1$!3QPTmF/H@WF-7$$!3qmmm\"G&R2:F1$!3NZL!y9:mg%F-7$$!3XLLLtK5F8F1$!3 pCxEC,D1[F-7$$!3_LLL$yP2D\"F1$!3K$)RC!>;u([F-7$$!3eLLL$HsV<\"F1$!3G'RB ?n*4O\\F-7$$!3!pmmT2Tb3\"F1$!3#R$yAF\\?$)\\F-7$$!3+-++]&)4n**F-$!3%oAy \"[G(***\\F-7$$!3yNL$e9XRd*F-$!3q<*3+Zk_*\\F-7$$!3cpmmTz\")\\F-7$$!3K.+]P$Qwy)F-$!3)f$3.(yJ&e\\F-7$$!37PLLL\\[%R)F-$!3wvo=& y$RC\\F-7$$!3qnmmT$!3\"e$Gi'*f^:))F\\v7$$!3gSnmmmr[R!#?$!3ub=!y4b'[RFbv7$$\" 3'oHLL3+TU)F\\v$\"3A;**3C$RZO)F\\v7$$\"3yELL$=2Vs\"F-$\"3G)zB5x>Xn\"F- 7$$\"3Khmmm7+#\\#F-$\"3GHn5:ZHYBF-7$$\"3)e*****\\`pfKF-$\"3#=Onq\"4gYH F-7$$\"3]imm\"*f#))3%F-$\"3VvJ(Hl^J]$F-7$$\"36HLLLm&z\"\\F-$\"3cVQ:VV9 gRF-7$$\"3;jmm;(HXx&F-$\"3[zTq%370L%F-7$$\"3>(******z-6j'F-$\"3BrK]%HV eg%F-7$$\"3W%*****\\C4puF-$\"3e#GJ^[:Wz%F-7$$\"3q\"******4#32$)F-$\"3W .Tfx<@:\\F-7$$\"3?#***\\P6[7()F-$\"3=A3y-8)G&\\F-7$$\"3q#****\\>()**\\F-7$$\"3K****\\FJ*G3\"F1$\"3+Dn+Xp=%)\\F-7$ $\"3G******H%=H<\"F1$\"3)[t-t_qq$\\F-7$$\"3qKLLo,\"QD\"F1$\"3%[%)*oP%p Z([F-7$$\"35mmm1>qM8F1$\"37SW!o0>')z%F-7$$\"3%)*******HSu]\"F1$\"3dR/r d;c1YF-7$$\"3'HLL$ep'Rm\"F1$\"3x#=\")f:F^T%F-7$$\"3')******R>4N=F1$\"3 Lf5e<(Q;?%F-7$$\"3#emm;@2h*>F1$\"3\\IK(RwtY+%F-7$$\"3]*****\\c9W;#F1$ \"3**y%Q_#\\W2QF-7$$\"3Lmmmmd'*GBF1$\"3w7+#3pn`i$F-7$$\"3j*****\\iN7]# F1$\"3U`#3=.UqW$F-7$$\"3aLLLt>:nEF1$\"3[VEYYCF-7$$\"\"%F*$\"3?)eqk " 0 "" {MPLTEXT 1 0 40 "xc := [sol ve(Df(x)=0)];\nyc := map(f,xc);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# xcG7$!\"\"\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ycG7$#!\"\"\"\" ##\"\"\"F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4 " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 257 "" 0 "" {TEXT 258 8 "Question" }{TEXT 308 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 63 "Find the equation of the tangent and normal lines to the graph " } {XPPEDIT 18 0 "y=x/(x+1)" "6#/%\"yG*&%\"xG\"\"\",&F&F'F'F'!\"\"" } {TEXT -1 20 " at the point where " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\" \"" }{TEXT -1 2 ". " }}{PARA 257 "" 0 "" {TEXT 269 8 "Solution" } {TEXT 309 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 21 "By the quotient rule, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = (``(1)*`. `*``(x+1)-``(x)*`.`*``(1))/((x+1)^2);" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&,& *(-%!G6#F&F&%\".GF&-F-6#,&%\"xGF&F&F&F&F&*(-F-6#F3F&F/F&-F-6#F&F&F(F&* $,&F3F&F&F&\"\"#F(" }{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " = " }{XPPEDIT 18 0 "1/((x+1)^2);" "6#*&\"\"\"F$*$,&%\"xGF$F$F$\"\"#!\" \"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 23 "The point on the curve " }{XPPEDIT 18 0 "y=x/(x+1)" "6# /%\"yG*&%\"xG\"\"\",&F&F'F'F'!\"\"" }{TEXT -1 6 " with " }{TEXT 362 1 "x" }{TEXT -1 16 " coordinate 1 is" }{XPPEDIT 18 0 " ``(1,1/2)" "6#-%! G6$\"\"\"*&F&F&\"\"#!\"\"" }{TEXT -1 56 " and the gradient of the tang ent line at this point is " }{XPPEDIT 18 0 "eval(dy/dx,x=1)=1/4" "6#/ -%%evalG6$*&%#dyG\"\"\"%#dxG!\"\"/%\"xGF)*&F)F)\"\"%F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 38 "The equation of this tangent line is : " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y-1/2=``(1/4)*( x-1)" "6#/,&%\"yG\"\"\"*&F&F&\"\"#!\"\"F)*&-%!G6#*&F&F&\"\"%F)F&,&%\"x GF&F&F)F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=x/4+1/4" "6#/%\"yG,&*&%\" xG\"\"\"\"\"%!\"\"F(*&F(F(F)F*F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 29 "The normal line has gradient " }{XPPEDIT 18 0 "-4" "6#,$ \"\"%!\"\"" }{TEXT -1 22 ", so its equation is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y-1/2=``(-4)*(x-1)" "6#/,&%\"yG\"\"\"*& F&F&\"\"#!\"\"F)*&-%!G6#,$\"\"%F)F&,&%\"xGF&F&F)F&" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y=-4*x+9/2" "6#/%\"yG,&*&\"\"%\"\"\"%\"xGF(!\"\"*&\"\"* F(\"\"#F*F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "f := x -> x/(1+x);\nDiff(f(x ),x);\nvalue(%);\nnormal(%);\neval(%,x=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*&9$\"\"\",&F- F.F.F.!\"\"F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%DiffG6$*&%\"xG \"\"\",&F'F(F(F(!\"\"F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&\"\"\"F %,&%\"xGF%F%F%!\"\"F%*&F'F%F&!\"#F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #*&\"\"\"F$*$),&%\"xGF$F$F$\"\"#F$!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"\"\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 245 "p1 := plot([x/(x+1),x/4+1/4,-4*x+9/2],x=-3 ..4,y=-2..4,discont=true,\n color=[red,blue,COLOR(RGB,0,.7 ,.2)]):\np2 := plot([[[-1,-2],[-1,4]],[[-3,1],[4,1]]],\n \+ color=black,linestyle=3):\nplots[display]([p1,p2],tickmarks=[3,4]);" } }{PARA 13 "" 1 "" {GLPLOT2D 406 320 320 {PLOTDATA 2 "6*-%'CURVESG6%7gn 7$$!\"$\"\"!$\"3++++++++:!#<7$$!3s\\uz\"p0k&HF-$\"3S;U[G996:F-7$$!3j4z 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$F($\"\"\"F*7$Fc`mFe[oF[[oF][o-%*AXESTICKSG6$F`[oFd`m-%+AXESLABELSG6%Q \"x6\"Q\"yF_\\o-%%FONTG6#%(DEFAULTG-%%VIEWG6$;F(Fc`m;FhjnFc`m" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q1" }}{PARA 0 "" 0 "" {TEXT -1 21 " Find the derivative " }{XPPEDIT 18 0 "`f '`(x) " "6#-%$f~'G6#%\"xG" }{TEXT -1 37 " for each of the following function s." }}{PARA 0 "" 0 "" {TEXT -1 10 " (a) " }{XPPEDIT 18 0 "f(x) = \+ (x^2-3*x+8)^4" "6#/-%\"fG6#%\"xG*$,(*$F'\"\"#\"\"\"*&\"\"$F,F'F,!\"\" \"\")F,\"\"%" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 11 " (b) " }{XPPEDIT 18 0 "f(x) = sqrt( x^2-4*x+2);" "6#/-%\"fG6#%\"xG-%%sqrtG6#,(*$F'\"\"#\"\"\"*&\"\"%F.F'F. !\"\"F-F." }{TEXT -1 4 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 " (c) " }{XPPEDIT 18 0 "f(x)=1/(x^3-5*x+2) ^5" "6#/-%\"fG6#%\"xG*&\"\"\"F)*$,(*$F'\"\"$F)*&\"\"&F)F'F)!\"\"\"\"#F )F/F0" }{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 " (d) " }{XPPEDIT 18 0 "f(x) = (x-2/x)^3;" "6#/- %\"fG6#%\"xG*$,&F'\"\"\"*&\"\"#F*F'!\"\"F-\"\"$" }{TEXT -1 3 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 11 " (e) \+ " }{XPPEDIT 18 0 "f(x)=3*x-sqrt(2*x+3)" "6#/-%\"fG6#%\"xG,&*&\"\"$\" \"\"F'F+F+-%%sqrtG6#,&*&\"\"#F+F'F+F+F*F+!\"\"" }{TEXT -1 2 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 " (f) \+ " }{XPPEDIT 18 0 "f(x)=sqrt(1+3*sqrt(4*x+1))" "6#/-%\"fG6#%\"xG-%%sqr tG6#,&\"\"\"F,*&\"\"$F,-F)6#,&*&\"\"%F,F'F,F,F,F,F,F," }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }{XPPEDIT 18 0 "`f '`(x) \+ = 4*(x^2-3*x+8)^3*(2*x-3);" "6#/-%$f~'G6#%\"xG*(\"\"%\"\"\"*$,(*$F'\" \"#F**&\"\"$F*F'F*!\"\"\"\")F*F0F*,&*&F.F*F'F*F*F0F1F*" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 4 "(b) " }{XPPEDIT 18 0 "`f '`(x) = (x-2) /sqrt(x^2-4*x+2);" "6#/-%$f~'G6#%\"xG*&,&F'\"\"\"\"\"#!\"\"F*-%%sqrtG6 #,(*$F'F+F**&\"\"%F*F'F*F,F+F*F," }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(c) " }{XPPEDIT 18 0 "`f '`(x) = -5*(3*x^2-5)/((x^3-5*x+2) ^6);" "6#/-%$f~'G6#%\"xG,$*(\"\"&\"\"\",&*&\"\"$F+*$F'\"\"#F+F+F*!\"\" F+*$,(*$F'F.F+*&F*F+F'F+F1F0F+\"\"'F1F1" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(d) " }{XPPEDIT 18 0 "`f '`(x) = 3*(x-2/x)^2*(1+2/(x^ 2));" "6#/-%$f~'G6#%\"xG*(\"\"$\"\"\"*$,&F'F**&\"\"#F*F'!\"\"F/F.F*,&F *F**&F.F**$F'F.F/F*F*" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 4 " (e) " }{XPPEDIT 18 0 "`f '`(x) = 3-1/sqrt(2*x+3);" "6#/-%$f~'G6#%\"xG, &\"\"$\"\"\"*&F*F*-%%sqrtG6#,&*&\"\"#F*F'F*F*F)F*!\"\"F2" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "(f) " }{XPPEDIT 18 0 "`f '`(x) = 3/( sqrt(1+3*sqrt(4*x+1))*sqrt(4*x+1));" "6#/-%$f~'G6#%\"xG*&\"\"$\"\"\"*& -%%sqrtG6#,&F*F**&F)F*-F-6#,&*&\"\"%F*F'F*F*F*F*F*F*F*-F-6#,&*&F5F*F'F *F*F*F*F*!\"\"" }{TEXT -1 4 " " }}}{PARA 0 "" 0 "" {TEXT -1 40 "___ _____________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 40 "__ ______________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q2" }} {PARA 0 "" 0 "" {TEXT -1 14 "Differentiate " }{XPPEDIT 18 0 "f(x) = u( x)*v(x);" "6#/-%\"fG6#%\"xG*&-%\"uG6#F'\"\"\"-%\"vG6#F'F," }{TEXT -1 17 " with respect to " }{TEXT 260 1 "x" }{TEXT -1 19 " in two ways, wh en " }{XPPEDIT 18 0 "u(x);" "6#-%\"uG6#%\"xG" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "v(x);" "6#-%\"vG6#%\"xG" }{TEXT -1 22 " are given as fo llows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " (a) " }{XPPEDIT 18 0 "u(x) = x^2+2*x+4,v(x) = x-2;" "6$/-%\"uG6#%\"xG ,(*$F'\"\"#\"\"\"*&F*F+F'F+F+\"\"%F+/-%\"vG6#F',&F'F+F*!\"\"" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "(b) " }{XPPEDIT 18 0 "u(x) = x+1/x,v(x) = sqrt(x);" "6$/-%\"uG6#% \"xG,&F'\"\"\"*&F)F)F'!\"\"F)/-%\"vG6#F'-%%sqrtG6#F'" }{TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "(c) \+ " }{XPPEDIT 18 0 "u(x) = x^2-5*x+2,u(x) = 1/x;" "6$/-%\"uG6#%\"xG,(*$F '\"\"#\"\"\"*&\"\"&F+F'F+!\"\"F*F+/-F%6#F'*&F+F+F'F." }{TEXT -1 2 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 5 "(a) " }{XPPEDIT 18 0 "`f '`(x) = 3*x^2;" "6#/-%$f~'G6#%\"xG*&\"\"$\"\"\"*$F'\"\"#F*" }{TEXT -1 9 " \+ (b) " }{XPPEDIT 18 0 "`f '`(x) = 3/2;" "6#/-%$f~'G6#%\"xG*&\"\"$\" \"\"\"\"#!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(x)-1/2" "6#,&-%%s qrtG6#%\"xG\"\"\"*&F(F(\"\"#!\"\"F+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "x ^(-3/2)" "6#)%\"xG,$*&\"\"$\"\"\"\"\"#!\"\"F*" }{TEXT -1 7 " (c) " } {XPPEDIT 18 0 "`f '`(x) = 1-2/(x^2);" "6#/-%$f~'G6#%\"xG,&\"\"\"F)*&\" \"#F)*$F'F+!\"\"F-" }{TEXT -1 1 " " }}}{PARA 0 "" 0 "" {TEXT -1 37 "__ ___________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 37 "_____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q3" }}{PARA 0 "" 0 "" {TEXT -1 48 "Find the derivativ es of the following functions:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 6 "(a) " }{XPPEDIT 18 0 "f(x) = (3*x-2)/(1- 2*x);" "6#/-%\"fG6#%\"xG*&,&*&\"\"$\"\"\"F'F,F,\"\"#!\"\"F,,&F,F,*&F-F ,F'F,F.F." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 6 "(b) " }{XPPEDIT 18 0 "f(x) = (x^2+6*x-4)/(2*x-9); " "6#/-%\"fG6#%\"xG*&,(*$F'\"\"#\"\"\"*&\"\"'F,F'F,F,\"\"%!\"\"F,,&*&F +F,F'F,F,\"\"*F0F0" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 6 "(c) " }{XPPEDIT 18 0 "f(t) = sqrt(t)/(5 *t+1);" "6#/-%\"fG6#%\"tG*&-%%sqrtG6#F'\"\"\",&*&\"\"&F,F'F,F,F,F,!\" \"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 5 "(a) " } {XPPEDIT 18 0 "`f '`(x) = -1/((1-2*x)^2);" "6#/-%$f~'G6#%\"xG,$*&\"\" \"F**$,&F*F**&\"\"#F*F'F*!\"\"F.F/F/" }{TEXT -1 7 " (b) " }{XPPEDIT 18 0 "`f '`(x) = (2*x^2-18*x-46)/((2*x-9)^2);" "6#/-%$f~'G6#%\"xG*&,(* &\"\"#\"\"\"*$F'F+F,F,*&\"#=F,F'F,!\"\"\"#YF0F,*$,&*&F+F,F'F,F,\"\"*F0 F+F0" }{TEXT -1 7 " (c) " }{XPPEDIT 18 0 "`f '`(t) = (1-5*t)/(2*sqrt (t)*(5*t+1)^2);" "6#/-%$f~'G6#%\"tG*&,&\"\"\"F**&\"\"&F*F'F*!\"\"F**( \"\"#F*-%%sqrtG6#F'F*,&*&F,F*F'F*F*F*F*F/F-" }{TEXT -1 2 " " }}} {PARA 0 "" 0 "" {TEXT -1 52 "_________________________________________ ___________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 52 "____________________________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }}{PARA 0 "" 0 "" {TEXT -1 48 "Find the derivatives of the following functions:" }}{PARA 0 "" 0 "" {TEXT -1 7 " (a) " } {XPPEDIT 18 0 "f(x)=(4*x^2-3)^3*(x^3-1)^6" "6#/-%\"fG6#%\"xG*&,&*&\"\" %\"\"\"*$F'\"\"#F,F,\"\"$!\"\"F/,&*$F'F/F,F,F0\"\"'" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 " (b) \+ " }{XPPEDIT 18 0 "g(t)=sqrt(1-t)/t" "6#/-%\"gG6#%\"tG*&-%%sqrtG6#,&\" \"\"F-F'!\"\"F-F'F." }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 7 " \+ (c) " }{XPPEDIT 18 0 "h(u) = ((u-2)/(2*u^2+1))^3;" "6#/-%\"hG6#%\"uG* $*&,&F'\"\"\"\"\"#!\"\"F+,&*&F,F+*$F'F,F+F+F+F+F-\"\"$" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 5 "(a) " }{XPPEDIT 18 0 "24*x*(4* x^2-3)^2*(x^3-1)^6+18*x^2*(4*x^2-3)^3*(x^3-1)^5 =6*x*(16*x^3-9*x-4)*(4 *x^2-3)^2*(x^3-1)^5" "6#/,&**\"#C\"\"\"%\"xGF',&*&\"\"%F'*$F(\"\"#F'F' \"\"$!\"\"F-,&*$F(F.F'F'F/\"\"'F'**\"#=F'*$F(F-F',&*&F+F'*$F(F-F'F'F.F /F.,&*$F(F.F'F'F/\"\"&F'*,F2F'F(F',(*&\"#;F'*$F(F.F'F'*&\"\"*F'F(F'F/F +F/F',&*&F+F'*$F(F-F'F'F.F/F-,&*$F(F.F'F'F/F;" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 " (b) " } {XPPEDIT 18 0 "`g '`(t) = (t-2)/(2*t^2*sqrt(1-t));" "6#/-%$g~'G6#%\"tG *&,&F'\"\"\"\"\"#!\"\"F**(F+F**$F'F+F*-%%sqrtG6#,&F*F*F'F,F*F," } {TEXT -1 9 " (c) " }{XPPEDIT 18 0 "`h '`(u) = -3*(u-2)^2*(2*u^2-8* u-1)/((2*u^2+1)^4);" "6#/-%$h~'G6#%\"uG,$**\"\"$\"\"\"*$,&F'F+\"\"#!\" \"F.F+,(*&F.F+*$F'F.F+F+*&\"\")F+F'F+F/F+F/F+*$,&*&F.F+*$F'F.F+F+F+F+ \"\"%F/F/" }{TEXT -1 1 " " }}}{PARA 0 "" 0 "" {TEXT -1 52 "___________ _________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 52 "_________________________________ ___________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q5" }}{PARA 0 "" 0 "" {TEXT -1 10 "(a) Given " }{XPPEDIT 18 0 "f(x)=sqrt(x^3-9*x^2+24*x)" "6#/-%\"fG6 #%\"xG-%%sqrtG6#,(*$F'\"\"$\"\"\"*&\"\"*F.*$F'\"\"#F.!\"\"*&\"#CF.F'F. F." }{TEXT -1 7 ", find " }{XPPEDIT 18 0 "`f '`(x)" "6#-%$f~'G6#%\"xG " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 24 "(b) Sketch the graph of " }{XPPEDIT 18 0 "y=f(x)" "6#/%\" yG-%\"fG6#%\"xG" }{TEXT -1 75 ", and find the coordinates of the maxim um and minimum points on the graph. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 6 " (a) " }{XPPEDIT 18 0 "`f '`(x) = (3*x^2-18*x+24)/(2*sqrt(x^3-9 *x^2+24*x));" "6#/-%$f~'G6#%\"xG*&,(*&\"\"$\"\"\"*$F'\"\"#F,F,*&\"#=F, F'F,!\"\"\"#CF,F,*&F.F,-%%sqrtG6#,(*$F'F+F,*&\"\"*F,*$F'F.F,F1*&F2F,F' F,F,F,F1" }{TEXT -1 6 ", (b) " }{XPPEDIT 18 0 "``(2,2*sqrt(5))" "6#-%! G6$\"\"#*&F&\"\"\"-%%sqrtG6#\"\"&F(" }{TEXT -1 2 ", " }{XPPEDIT 18 0 " ``(4,4)" "6#-%!G6$\"\"%F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 231 "f := x -> sqrt(x^ 3-9*x^2+24*x):\n'f(x)'=f(x);\np1 := plot(f(x),x=0..7.9):\nt1 := plots[ textplot]([[7.9,-.25,`x`],[-.23,11.3,`y`]]):\nplots[display]([p1,t1],t ickmarks=[3,4],font=[HELVETICA,8],labels=[``,``],view=[-.23..7.9,-.25. .11.3]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"fG6#%\"xG*$,(*$)F'\" \"$\"\"\"F-*&\"\"*F-)F'\"\"#F-!\"\"*&\"#CF-F'F-F-#F-F1" }}{PARA 13 "" 1 "" {GLPLOT2D 273 237 237 {PLOTDATA 2 "6)-%'CURVESG6$7W7$$\"\"!F)F(7$ $\"3Mmm\"z%z$\\I%!#>$\"3]'))*H?(e#35!#<7$$\"3oKL$e*e()4')F-$\"3iD;x=mJ 99F07$$\"3-+]P%Q\"[\"H\"!#=$\"3];(zW-@!=s\"F9$\"3-!R$ *pD![n>F07$$\"3)**\\(=xe6rCF9$\"3;P=n&>qHK#F07$$\"3TL$3_dc-A$F9$\"3]>, Ta\"zJh#F07$$\"3hmmT02B0\\F9$\"3$e*fF0$\"35Hi]Er$> Z%F07$$\"3#****\\inUo9#F0$\"3]Dry!H__Y%F07$$\"3=++DV9s6BF0$\"3$=#p]*fM GW%F07$$\"3YmTNfrWhCF0$\"3yx1&fY(G6WF07$$\"3;LLe%F07$$\"3'**\\(=$\\G9H$F0$\"3(en)fv? 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 1 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx = (x^2+ 1)/(2*x^2*sqrt(x-1/x));" "6#/*&%#dyG\"\"\"%#dxG!\"\"*&,&*$%\"xG\"\"#F& F&F&F&*(F-F&*$F,F-F&-%%sqrtG6#,&F,F&*&F&F&F,F(F(F&F(" }{TEXT -1 5 " , \+ " }{XPPEDIT 18 0 "y-sqrt(3/2)=5/8" "6#/,&%\"yG\"\"\"-%%sqrtG6#*&\"\" $F&\"\"#!\"\"F-*&\"\"&F&\"\")F-" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt( 2/3)*(x-2)" "6#*&-%%sqrtG6#*&\"\"#\"\"\"\"\"$!\"\"F),&%\"xGF)F(F+F)" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 399 "f := x -> sqrt(x-1/x):\ng := x -> sqrt(3/2)+5/8 *sqrt(2/3)*(x-2):\np1 := plot(f(x),x=1..4.9,numpoints=100,thickness=1) :\np2 := plot(f(x),x=-1..-0.1,numpoints=100,thickness=1):\np3 := plot( g(x),x=-1.5..4.9,color=COLOR(RGB,0,.7,0),thickness=2):\nt1 := plots[te xtplot]([[4.9,-.08,`x`],[.2,2.7,`y`]]):\nplots[display]([p||(1..3),t1] ,tickmarks=[3,4],font=[HELVETICA,8],labels=[``,``],view=[-2.3..4.9,-.5 ..2.7]);" }}{PARA 13 "" 1 "" {GLPLOT2D 430 213 213 {PLOTDATA 2 "6+-%'C URVESG6%7bq7$$\"\"\"\"\"!$F*F*7$$\"3i[[[=#31-\"!#<$\"3*HBmnv3*>?!#=7$$ \"3,(pppV;7/\"F/$\"36qfV-ubUGF27$$\"3C==oPv9f5F/$\"3M3+(e@Z5R$F27$$\"3 YRRRQ'yq2\"F/$\"3?)eZ\\*HSbQF27$$\"3dgggM%4u6\"F/$\"3IMyalJ!or%F27$$\" 3]^^^pv+e6F/$\"3%>T6U2ujU&F27$$\"3iddd[FT)>\"F/$\"3635\\U)RI.'F27$$\"3 \"*yyy@N(eB\"F/$\"3YLW&[c^C`'F27$$\"3mjjj#)>mu7F/$\"3:*3Ch'=+,qF27$$\" 3TLLLkqx98F/$\"3[J!f^E1WW(F27$$\"3UXXX'\\jZN\"F/$\"3++X$=)=c_yF27$$\"3 +&[[oX%*eR\"F/$\"3EE:_>qAV#)F27$$\"3A:::HE7K9F/$\"3%p_36e`lc)F27$$\"3s aaaqp!HZ\"F/$\"3WHq>=8a5*)F27$$\"3UOOOw(eQ^\"F/$\"3@xE`&y(RP#*F27$$\"3 ]XXXHMK`:F/$\"3Yck/lP*p`*F27$$\"3KCCC:6;*e\"F/$\"31]/#H'*Ruz*F27$$\"3) RRR**eva]=.p$36F/7$$\"3pjjj/ilE=F/$\"3'3wFbp?58\"F/7$$ \"3I...`eF/$\"3#[WQ\"z#GW<\"F/7$$ \"3eUUU77_W>F/$\"3eFZeDI$f>\"F/7$$\"31111(*3@')>F/$\"3k%\\X1*zn<7F/7$$ \"33444!*=]A?F/$\"31CSyr\"\\hB\"F/7$$\"3Czyycrph?F/$\"3mX#\\F<^cD\"F/7 $$\"3zaaa-(*=-@F/$\"3+@@1D7Mv7F/7$$\"3ZssseP!=9#F/$\"3aoLR.A=%H\"F/7$$ \"3A+++&RK,=#F/$\"3d/w,f$Q?J\"F/7$$\"3zsss5)*oAAF/$\"35>L/H&f9L\"F/7$$ \"3adddT&H4E#F/$\"3')zN@Lsc[8F/7$$\"3aOOO;'evy.0$F/$\"3sy\"HhL:+l\"F/7$$\"3qyyy;xE'3$F/$\"3;6LGLB+i;F/7$$\"3maa ae59FJF/$\"3WP/,wy^v;F/7$$\"3=&[[))oAm;$F/$\"39*)=)=wR%)o\"F/7$$\"3*fd dn_yg?$F/$\"3\\40eCjA,==3: *)GO$F/$\"3Qp1p*Hk3v\"F/7$$\"39&[[o[RQS$F/$\"3EJ&peSMNw\"F/7$$\"39111- %44W$F/$\"3ccmkC_!\\x\"F/7$$\"3/%RRzfy=[$F/$\"3WGn6_$F/$\"3AA*G^l*>*z\"F/7$$\"3oUUU%yl-c$F/$\"3QQN\"e-34\"=F/7$$\"3O# ===w`6g$F/$\"35$*3Q&HPI#=F/7$$\"31nmm\"38)QOF/$\"33eQ=^07M=F/7$$\"3k[[ [w8QxOF/$\"3KC`-)z&QX=F/7$$\"3!GFFnNl*>PF/$\"3s/E%[$fsd=F/7$$\"3!)yyyi ;_ePF/$\"3_zj-_B\")o=F/7$$\"3Atss2F&zz$F/$\"3'>^zYFn+)=F/7$$\"37wvvkI0 QQF/$\"35<#=w;H9*=F/7$$\"3Y:::'48\\(QF/$\"3L'*RI#))*z,>F/7$$\"3a+++'p/ U\"RF/$\"3-iXsz$zF\">F/7$$\"3yjjj0s>`RF/$\"3Znr:9,gB>F/7$$\"3)411122\\ *RF/$\"3\\&3[$))Q4N>F/7$$\"3gddd>UrJSF/$\"3[$=#=q&o^%>F/7$$\"3ymmmG?=u SF/$\"3E!>`65:n&>F/7$$\"35CCC_f]7TF/$\"3%Q13g7lq'>F/7$$\"3+VUUraU]TF/$ \"3/]4&*)4Us(>F/7$$\"3eIIIO6?\">%F/$\"3]^><1i6))>F/7$$\"3:;::Z<9KUF/$ \"3mGQrAM'*)*>F/7$$\"35111T;RpUF/$\"3!z<5)>Bx3?F/7$$\"39wvvt$\\*3VF/$ \"3W?fl)4F\">?F/7$$\"3mnmmaQbZVF/$\"3R0j;5@:S*Q%F/$\"3 %eb$)[5&**R?F/7$$\"3l+++%Q6cU%F/$\"311AW7dI\\?F/7$$\"3W444I(*>nWF/$\"3 /.xy6$Q*f?F/7$$\"39zyyh\\H1XF/$\"37$H(\\d^()p?F/7$$\"3`===wD,XXF/$\"3) )QDW98mz?F/7$$\"3]sss!*4w$e%F/$\"3q9jX:GBYF/$\"3-. 0z/@G*4#F/7$$\"3SkjjWM>@F/7$$\"36hgg\"*)*\\TZF/$\"3$4$4*>r>&G@F/7$$\"3a)yy[\\J?y%F/$\"3@ V04&fP%Q@F/7$$\"3CRRRk[ZA[F/$\"3cwekz7G[@F/7$$\"3)\\UU->v(e[F/$\"3M`0L aD2d@F/7$$\"3M+++++++\\F/$\"3a,q-wn+n@F/-%'COLOURG6&%$RGBG$\"#5!\"\"F+ F+-%*THICKNESSG6#F)-F$6%7`q7$$Fg[mF*F+7$$!3,XXXX^)[!**F2$\"3%\\Lg]/XDQ \"F27$$!3m344>i7A)*F2$\"3uF954rk%*=F27$$!3d!444:b!H(*F2$\"3A<$GV_.SM#F 27$$!3eFFFZrON'*F2$\"3YiF,s5#fs#F27$$!3BOOOEW7U&*F2$\"3'yKui#\\AiIF27$ $!3%y\"==)[wcX*F2$\"3)pr%ytBhYLF27$$!3CXXXbY;m$*F2$\"3iZQux9'fJmNTF27$ $!3;FFF2uR'3*F2$\"31u]ptFr!Q%F27$$!3CtssKRz-!*F2$\"3cV/Jy])ye%F27$$!3U \"===Aw'3*)F2$\"3Z6OU2H$G\"[F27$$!3%\\XXX0sT\"))F2$\"3>HSqXj4J]F27$$!3 f<==y(*4B()F2$\"3,Aq8RQ=N_F27$$!3GOOOEuRS')F2$\"3!Gmw;2^eT&F27$$!3L\"4 44Lc?a)F2$\"3B+LAe.fDcF27$$!3ajjjB&[(e%)F2$\"3[WH*)zMU*z&F27$$!3W444R< %=O)F2$\"3OCzzeYq(*fF27$$!37baa9G1w#)F2$\"35jV9X32qhF27$$!31444R(\\>=) F2$\"31nN6)=fhN'F27$$!3#[aaa$=L#4)F2$\"3hG/:qh!eZqF27$$!3g344*pGTs(F2$\"3eD:\"[*)elA(F27$$!33kjj`-QSwF2$\"3I; 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Sketch the graph of " }{XPPEDIT 18 0 "y=f(x)" "6#/%\"y G-%\"fG6#%\"xG" }{TEXT -1 75 ", and find the coordinates of the maximu m and minimum points on the graph. " }}{PARA 0 "" 0 "" {TEXT -1 52 "__ __________________________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 52 "____________________ ________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }} }{SECT 1 {PARA 0 "" 0 "" {TEXT -1 25 "Code for drawing pictures" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 " " {TEXT -1 23 " 1st chain rule picture" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 686 "p1 := plot([[[-9,0],[-6 ,0]],[[-2,0],[2,0]],[[6,0],[9,0]],\n [[-8.2,0.2],[-7.8,0],[-8.2,-0.2] ],[[7.8,0.2],[8.2,0],[7.8,-0.2]],\n [[-0.2,0.2],[0.2,0],[-0.2,-0.2]]] ,color=black):\np2 := plots[polygonplot]([[[2,2],[6,2],[6,-2],[2,-2]], \n [[-2,2],[-2,-2],[-6,-2],[-6,2]]],color=COLOR(RGB,0.7,0.7,0.9)):\np 3 := plot([[-7,3],[7,3],[7,-3],[-7,-3],[-7,3]],color=red):\nt1 := plot s[textplot]([[4,.3,`take the`],[4,-.3,`square root`],\n [-4,.3,`squar e and`],[-4,-.3,`subtract from 1`]],font=[HELVETICA,9]):\nt2 := plots[ textplot]([[-9.5,0,`x`],[0,.7,`u`],[9.5,0,`y`]],font=[HELVETICA,9]):\n t3 := plots[textplot]([0,-3.4,`f`],color=red,font=[HELVETICA,9]):\nplo ts[display]([p1,p2,p3,t1,t2,t3],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 22 "2nd chain rule picture" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 596 "p1 := plo t([[[-9,0],[-6,0]],[[-2,0],[2,0]],[[6,0],[9,0]],\n [[-8.2,0.2],[-7.8,0 ],[-8.2,-0.2]],[[7.8,0.2],[8.2,0],[7.8,-0.2]],\n [[-0.2,0.2],[0.2,0],[ -0.2,-0.2]]],color=black):\np2 := plots[polygonplot]([[[2,2],[6,2],[6, -2],[2,-2]],\n [[-2,2],[-2,-2],[-6,-2],[-6,2]]],color=COLOR(RGB,0.7, 0.7,0.9)):\np3 := plot([[-7,3],[7,3],[7,-3],[-7,-3],[-7,3]],color=red) :\nt1 := plots[textplot]([[4,0,`g`],[-10,0,`x`],[0,1,`u = u(x)`],\n \+ [-4,0,`u`],[12.5,-.3,`y = g(u(x)) = f(x)`]],font=[HELVETICA,9]):\nt2 : = plots[textplot]([0,-3.4,`f`],color=red,font=[HELVETICA,9]):\nplots[d isplay]([p1,p2,p3,t1,t2],axes=none);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }