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{SECT 0 {SECT 1 {PARA 256 "" 0 "" {TEXT -1 27 "Chapter 9\n\nDifferenti
ation\n" }}{PARA 0 "" 0 "" {TEXT 269 31 "\251 Copyright 2003 by Andr
\351 Heck." }}}{SECT 1 {PARA 0 "" 0 "" {TEXT 257 1 "1" }{TEXT -1 6 ". \+
Let " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "g
" "6#%\"gG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "h" "6#%\"hG" }{TEXT 
-1 60 " be the following multivariate real functions\n              " 
}{XPPEDIT 18 0 "f(x,y):=ln(1+x^4+y^4)/sqrt(x^2+y^2)" "6#>-%\"fG6$%\"xG
%\"yG*&-%#lnG6#,(\"\"\"F.*$F'\"\"%F.*$F(F0F.F.-%%sqrtG6#,&*$F'\"\"#F.*
$F(F7F.!\"\"" }{TEXT -1 11 "\n\n         " }{XPPEDIT 18 0 "g(x,y,z):=1
/sqrt((x-a)^2+(y-b)^2+(z-c)^2)" "6#>-%\"gG6%%\"xG%\"yG%\"zG*&\"\"\"F+-
%%sqrtG6#,(*$,&F'F+%\"aG!\"\"\"\"#F+*$,&F(F+%\"bGF3F4F+*$,&F)F+%\"cGF3
F4F+F3" }{TEXT -1 11 "\n\n         " }{XPPEDIT 18 0 "h(x,y,z):=z/(x^2+
y^2+z^2)" "6#>-%\"hG6%%\"xG%\"yG%\"zG*&F)\"\"\",(*$F'\"\"#F+*$F(F.F+*$
F)F.F+!\"\"" }{TEXT -1 2 "\n\n" }{TEXT 263 3 "(a)" }{TEXT -1 38 " Dete
rmine all partial derivatives of " }{XPPEDIT 18 0 "f" "6#%\"fG" }
{TEXT -1 14 " of order 2.\n\n" }{TEXT 264 3 "(b)" }{TEXT -1 12 " Check
 that " }{XPPEDIT 18 0 "g" "6#%\"gG" }{TEXT -1 82 " is a solution of t
he Laplace diffeential equation, i.e., \n                       " }
{XPPEDIT 18 0 "(diff(``, x,x)+diff(``, y,y)+diff(``,z,z))*g=0" "6#/*&,
(-%%diffG6%%!G%\"xGF*\"\"\"-F'6%F)%\"yGF.F+-F'6%F)%\"zGF1F+F+%\"gGF+\"
\"!" }{TEXT -1 3 ".\n\n" }{TEXT 265 3 "(c)" }{TEXT -1 12 " Check that \+
" }{XPPEDIT 18 0 "h" "6#%\"hG" }{TEXT -1 71 " is a solution of the dif
ferenial equation\n                            " }{XPPEDIT 18 0 "diff(
h,y,x)+(4*x/(x^2+y^2+z^2)*diff(h,y))=0" "6#/,&-%%diffG6%%\"hG%\"yG%\"x
G\"\"\"**\"\"%F+F*F+,(*$F*\"\"#F+*$F)F0F+*$%\"zGF0F+!\"\"-F&6$F(F)F+F+
\"\"!" }{TEXT -1 2 ".\n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 44 "f := (x,y) -> ln(1+x^4+y^4) / sqrt(x^2+y^2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"xG%\"yG6\"6$%)operatorG%&
arrowGF)*&-%#lnG6#,(\"\"\"F2*$)9$\"\"%F2F2*$)9%F6F2F2F2-%%sqrtG6#,&*$)
F5\"\"#F2F2*$)F9F@F2F2!\"\"F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 48 "g := (x,y,z) -> 1/sqrt((x-a)^2+(y-b)^2+(z-c)^2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6%%\"xG%\"yG%\"zG6\"6$%)operat
orG%&arrowGF**&\"\"\"F/-%%sqrtG6#,(*$),&9$F/%\"aG!\"\"\"\"#F/F/*$),&9%
F/%\"bGF9F:F/F/*$),&9&F/%\"cGF9F:F/F/F9F*F*F*" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 32 "h := (x,y,z) -> z/(x^2+y^2+z^2);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"hGf*6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF
**&9&\"\"\",(*$)9$\"\"#F0F0*$)9%F5F0F0*$)F/F5F0F0!\"\"F*F*F*" }}}
{SECT 0 {PARA 0 "" 0 "" {TEXT 266 3 "(a)" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "diff(f(x,y), x$2);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6
#,,**\"#7\"\"\"%\"xG\"\"#,(F&F&*$)F'\"\"%F&F&*$)%\"yGF,F&F&!\"\",&*$)F
'F(F&F&*$)F/F(F&F&#F0F(F&**\"#;F&F'\"\"'F)!\"#F1F6F0**\"\")F&F'F,F)F0F
1#!\"$F(F0**\"\"$F&-%#lnG6#F)F&F1#!\"&F(F'F(F&*&FAF&F1F=F0" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "diff(f(x,y), x,y);" }}{PARA 12 "" 
1 "" {XPPMATH 20 "6#,**,\"#;\"\"\"%\"xG\"\"$,(F&F&*$)F'\"\"%F&F&*$)%\"
yGF,F&F&!\"#,&*$)F'\"\"#F&F&*$)F/F4F&F&#!\"\"F4F/F(F8*,F,F&F'F(F)F8F1#
!\"$F4F/F&F8*,F,F&F/F(F)F8F1F:F'F&F8*,F(F&-%#lnG6#F)F&F1#!\"&F4F'F&F/F
&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "diff(f(x,y), y$2);" 
}}{PARA 12 "" 1 "" {XPPMATH 20 "6#,,**\"#7\"\"\"%\"yG\"\"#,(F&F&*$)%\"
xG\"\"%F&F&*$)F'F-F&F&!\"\",&*$)F,F(F&F&*$)F'F(F&F&#F0F(F&**\"#;F&F'\"
\"'F)!\"#F1F6F0**\"\")F&F'F-F)F0F1#!\"$F(F0**\"\"$F&-%#lnG6#F)F&F1#!\"
&F(F'F(F&*&FAF&F1F=F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}}{SECT 0 {PARA 0 "" 0 "" {TEXT 267 3 "(b)" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 9 "g(x,y,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&\"\"
\"F$*$,4*$)%\"xG\"\"#F$F$*(F*F$F)F$%\"aGF$!\"\"*$)F,F*F$F$*$)%\"yGF*F$
F$*(F*F$F2F$%\"bGF$F-*$)F4F*F$F$*$)%\"zGF*F$F$*(F*F$F9F$%\"cGF$F-*$)F;
F*F$F$#F$F*F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "radnormal(
 diff(%,x$2) + diff(%,y$2) + diff(%,z$2) );" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}}{SECT 0 {PARA 0 "" 0 "" {TEXT 268 3 "(c)" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 9 "h(x,y,z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&%\"zG
\"\"\",(*$)%\"xG\"\"#F%F%*$)%\"yGF*F%F%*$)F$F*F%F%!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "normal( diff(%,x,y) + 4*x/(x^2+y^2+
z^2)*diff(%,y) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 0 "" 0 "" 
{TEXT 258 2 "2." }{TEXT -1 53 " Compare the result of the following Ma
ple commands:\n" }{TEXT 0 60 "   > diff(f(x), x);\n   > convert(%, D);
\n   > unapply(%, x);\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "re
start;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "diff(f(x), x);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%%diffG6$-%\"fG6#%\"xGF)" }}}{PARA 
0 "" 0 "" {TEXT -1 77 "Here, we differentiated a formula and got a new
 formula, i.e., the derivative" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 14 "convert(%, D);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#--%\"DG6#%\"fG
6#%\"xG" }}}{PARA 0 "" 0 "" {TEXT -1 25 "We changed the notation: " }
{XPPEDIT 18 0 "D(f)" "6#-%\"DG6#%\"fG" }{TEXT -1 35 " is the derivativ
e of the function " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT -1 61 ". The d
erivative is a function and we computed  its value in " }{XPPEDIT 18 
0 "x" "6#%\"xG" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 15 "unapply(%, x );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"DG6#%\"fG
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 37 "Finally, we converted the formula in " }{XPPEDIT 18 0 "x
" "6#%\"xG" }{TEXT -1 30 " into a function depending on " }{XPPEDIT 
18 0 "x" "6#%\"xG" }{TEXT -1 1 "." }}}{SECT 1 {PARA 0 "" 0 "" {TEXT 
259 2 "3." }{TEXT -1 40 " Compute the derivative of the function " }
{XPPEDIT 18 0 "f(x) := max(x^3,x)" "6#>-%\"fG6#%\"xG-%$maxG6$*$F'\"\"$
F'" }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "resta
rt;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f := x-> max(x^3,x);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&a
rrowGF(-%$maxG6$*$)9$\"\"$\"\"\"F1F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "diff(f(x),x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*P
IECEWISEG6)7$\"\"\"2%\"xG!\"\"7$%*undefinedG/F)F*7$,$*&\"\"$F')F)\"\"#
F'F'2F)\"\"!7$F,/F)F57$F'2F)F'7$F,/F)F'7$F/2F'F)" }}}{PARA 0 "" 0 "" 
{TEXT -1 67 "This answer becomes more clear when we first consider the
 function " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT -1 33 " as a piecewise
 defined function." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "conver
t(f(x), piecewise);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%*PIECEWISEG6&
7$%\"xG1F'!\"\"7$*$)F'\"\"$\"\"\"2F'\"\"!7$F'1F'F.7$F+2F.F'" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "diff(%,x);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#-%*PIECEWISEG6)7$\"\"\"2%\"xG!\"\"7$%*undefinedG/F)F
*7$,$*&\"\"$F')F)\"\"#F'F'2F)\"\"!7$F,/F)F57$F'2F)F'7$F,/F)F'7$F/2F'F)
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "
" 0 "" {TEXT 260 2 "4." }{TEXT -1 18 " Let the function " }{XPPEDIT 
18 0 "y(x)" "6#-%\"yG6#%\"xG" }{TEXT -1 26 " be implicitly defined by \+
" }{XPPEDIT 18 0 "sqrt(x)+sqrt(y)=1" "6#/,&-%%sqrtG6#%\"xG\"\"\"-F&6#%
\"yGF)F)" }{TEXT -1 25 ". Compute the derivative " }{XPPEDIT 18 0 "`y'
`" "6#%#y'G" }{TEXT -1 27 " and the second derivative " }{XPPEDIT 18 
0 "`y''`" "6#%$y''G" }{TEXT -1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 
"eq := sqrt(x) + sqrt(y) = 1 ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#e
qG/,&*$%\"xG#\"\"\"\"\"#F**$%\"yGF)F*F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "implicitdiff(eq, y ,x);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#,$*&%\"yG#\"\"\"\"\"#%\"xG#!\"\"F(F+" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 25 "implicitdiff(eq, y, x$2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,$*(\"\"#!\"\",&*&%\"yG#\"\"\"F%%\"xGF+F+*$)F,#\"\"$F%F
+F+F+F,#!\"&F%F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 0 "" 0 "" {TEXT 261 2 "5." }{TEXT -1 28 " Let the bivari
ate function " }{XPPEDIT 18 0 "z(x,y)" "6#-%\"zG6$%\"xG%\"yG" }{TEXT 
-1 26 " be implicitly defined by " }{XPPEDIT 18 0 "h(x,y,z)=0" "6#/-%
\"hG6%%\"xG%\"yG%\"zG\"\"!" }{TEXT -1 31 ", for some trivariate functi
on " }{XPPEDIT 18 0 "h" "6#%\"hG" }{TEXT -1 25 ". Determine formulae f
or " }{XPPEDIT 18 0 "diff(z,x)" "6#-%%diffG6$%\"zG%\"xG" }{TEXT -1 5 "
 and " }{XPPEDIT 18 0 "diff(z,y,x)" "6#-%%diffG6%%\"zG%\"yG%\"xG" }
{TEXT -1 26 ". What are the result for " }{XPPEDIT 18 0 "h=sqrt(x)+sqr
t(y)+sqrt(z)-1" "6#/%\"hG,*-%%sqrtG6#%\"xG\"\"\"-F'6#%\"yGF*-F'6#%\"zG
F*F*!\"\"" }{TEXT -1 2 "?\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "implicitdiff
(h(x,y,z),z,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&--&%\"DG6#\"\"
\"6#%\"hG6%%\"xG%\"yG%\"zGF*--&F(6#\"\"$F+F-!\"\"F6" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 25 "dzdx := convert(%, diff);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%%dzdxG,$*&-%%diffG6$-%\"hG6%%\"xG%\"yG%\"zGF-\"
\"\"-F(6$F*F/!\"\"F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "d2z
dxdy := convert(implicitdiff(h(x,y,z),z,x,y), diff);" }}{PARA 12 "" 1 
"" {XPPMATH 20 "6#>%(d2zdxdyG*&,**&-%%diffG6%-%\"hG6%%\"xG%\"yG%\"zGF.
F/\"\"\")-F)6$F+F0\"\"#F1!\"\"*(-F)6%F+F.F0F1-F)6$F+F/F1F3F1F1*(-F)6$F
+F.F1-F)6%F+F/F0F1F3F1F1*(F=F1-F)6$F+-%\"$G6$F0F5F1F:F1F6F1F3!\"$" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "h := (x,y,z) -> sqrt(x) + sq
rt(y) + sqrt(z) - 1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hGf*6%%\"x
G%\"yG%\"zG6\"6$%)operatorG%&arrowGF*,*-%%sqrtG6#9$\"\"\"-F06#9%F3-F06
#9&F3F3!\"\"F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "dzdx;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&%\"xG#!\"\"\"\"#%\"zG#\"\"\"F(
F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "d2zdxdy;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#,$*&\"\"\"F%*(\"\"#F%%\"xG#F%F'%\"yG#F%F'!\"\"F%
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "
" 0 "" {TEXT 262 2 "6." }{TEXT -1 23 " Consider the function " }
{XPPEDIT 18 0 "f[n]" "6#&%\"fG6#%\"nG" }{TEXT -1 47 " recursively defi
ned by\n                       " }{XPPEDIT 18 0 "f[0]=0, f[1]=x, f[n]=
f[n-1]+sin(f[n-2])" "6%/&%\"fG6#\"\"!F'/&F%6#\"\"\"%\"xG/&F%6#%\"nG,&&
F%6#,&F0F+F+!\"\"F+-%$sinG6#&F%6#,&F0F+\"\"#F5F+" }{TEXT -1 5 " for " 
}{XPPEDIT 18 0 "n*`>`*1" "6#*(%\"nG\"\"\"%\">GF%F%F%" }{TEXT -1 90 ".
\nDetermine (by automatic differentation) a procedure to compute the f
irst derivative of  " }{XPPEDIT 18 0 "f[n]" "6#&%\"fG6#%\"nG" }{TEXT 
-1 2 ".\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}
{PARA 0 "" 0 "" {TEXT -1 38 "You may be tempted to do the following" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "f := proc(n,x) \n  if n=0 \+
then\n    0\n  elif n=1 then \n    x\n  else \n    f(n-1,x) + sin(f(n-
2,x))\n  end if\nend proc;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*
6$%\"nG%\"xG6\"F)F)@'/9$\"\"!F-/F,\"\"\"9%,&-F$6$,&F,F/F/!\"\"F0F/-%$s
inG6#-F$6$,&F,F/\"\"#F5F0F/F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "f(6,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*%\"xG\"\"
\"*&\"\"#F%-%$sinG6#F$F%F%-F)6#,&F$F%F(F%F%-F)6#,&F$F%*&F'F%F(F%F%F%" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "fp := D[2](f);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#fpG-&%\"DG6#\"\"#6#%\"fG" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "fp(6,x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#--&%\"DG6#\"\"#6#%\"fG6$\"\"'%\"xG" }}}{PARA 0 "" 0 "" 
{TEXT -1 332 "The problem is that automatic differentiation is only de
fined for iteratively defined functions in Maple. So, we redefine the \+
function definition. To understand the way we define the function iter
atively, first verify that the function can also be recurisively defin
ed as follows:\n                                                   " }
{XPPEDIT 18 0 "g[1]=0, f[1]=x, f[n]=f[n-1]+sin(g[n-1]) ,g[n]=f[n-1]" "
6&/&%\"gG6#\"\"\"\"\"!/&%\"fG6#F'%\"xG/&F+6#%\"nG,&&F+6#,&F1F'F'!\"\"F
'-%$sinG6#&F%6#,&F1F'F'F6F'/&F%6#F1&F+6#,&F1F'F'F6" }{TEXT -1 40 ".\nT
he iterative version of the function " }{XPPEDIT 18 0 "f" "6#%\"fG" }
{TEXT -1 19 " can be as follows:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 152 "f := proc(n,x)\n  local i, G, F, TMP;\n  G := 0;\n  F := x;\n
  for i from 2 to n do \n    TMP := F;\n    F := F + sin(G);\n    G :=
 TMP\n  end do;\n  F\nend proc;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%
\"fGf*6$%\"nG%\"xG6&%\"iG%\"GG%\"FG%$TMPG6\"F.C&>8%\"\"!>8&9%?(8$\"\"#
\"\"\"9$%%trueGC%>8'F4>F4,&F4F9-%$sinG6#F1F9>F1F>F4F.F.F." }}}{PARA 0 
"" 0 "" {TEXT -1 11 "An example:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 7 "f(6,x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,*%\"xG\"\"\"*&\"\"#
F%-%$sinG6#F$F%F%-F)6#,&F$F%F(F%F%-F)6#,&F$F%*&F'F%F(F%F%F%" }}}{PARA 
0 "" 0 "" {TEXT -1 44 "A procedure to compute the first derivative:" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "fp := D[2](f);" }}{PARA 12 
"" 1 "" {XPPMATH 20 "6#>%#fpGf*6$%\"nG%\"xG6)%\"iG%\"GG%\"FG%$TMPG%#Gx
G%#FxG%%TMPxG6\"F1C(>8(\"\"!>8%F5>8)\"\"\">8&9%?(8$\"\"#F:9$%%trueGC(>
8*F9>8'F<>F9,&F9F:*&-%$cosG6#F7F:F4F:F:>F<,&F<F:-%$sinGFMF:>F4FE>F7FGF
9F1F1F1" }}}{PARA 0 "" 0 "" {TEXT -1 25 "Let us check the example:" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "fp(6,x);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,*\"\"\"F$*&\"\"#F$-%$cosG6#%\"xGF$F$*&-F(6#,&F*F$-%$si
nGF)F$F$,&F$F$F'F$F$F$*&-F(6#,&F*F$*&F&F$F/F$F$F$,&F$F$*&F&F$F'F$F$F$F
$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "testeq( fp(6,x) = diff
(f(6,x),x) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "0" 0 }{VIEWOPTS 1 1 0 
1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }