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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 10 "Chain Rule" }}{PARA 3 "" 
0 "" {TEXT -1 14 "THE CHAIN RULE" }}{PARA 0 "" 0 "" {TEXT -1 53 "EXAMP
LE:  Use Maple V to illustrate the chain rule in" }}{PARA 0 "" 0 "" 
{TEXT -1 48 "                   differentiating  sqrt(x^2+5)." }}
{PARA 0 "" 0 "" {TEXT -1 71 "SOLUTION:   Here we define  f(z)= sqrt(z)
,  and  g(x)=x^2+5,  so that  " }}{PARA 0 "" 0 "" {TEXT -1 43 "       \+
              f(g(x)) = sqrt(x^2+5)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "f := x->sqrt(x);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "g := x -> x^2+5;" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 9 "(f@g)(x);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 67 "By the chain rule the derivative can be evaluated as the \+
product of" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 " D(f)(z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "with
  z=g(x)  and" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "D(g)(x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "which
 can be computed by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 19 "D(f)(g(x))*D(g)(x);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "Of course Maple V permits
 the direct calculation of the derivative." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "diff(sqrt(x^2+5),x);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "EXAMPLE: page 251, number 29" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "f:=x->(sin(x)^2)*(tan(x)^4)/
((x^2+1))^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "g:=x->ln(f(
x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "g(x);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "h:=x->2*ln(sin(x))+4*ln(tan(x))-2*l
n(x^2+1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "plot(\{g(x),h(
x)\},x=-5..5,y=-10..10);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Note.
 The graph above is to show the functions g(x) and h(x) are the same.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "diff(g(x),x);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "diff(h(x),x);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "k:=x->f(x)*diff(h(x),x);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "k(x);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 20 "f1:=x->diff(f(x),x);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 6 "f1(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
36 "plot(\{f(x),k(x)\},x=-5..5,y=-10..10);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 37 "plot(\{f1(x),k(x)\},x=-5..5,y=-10..10);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 66 "The graph above is to show the functions \+
f1 and k(x) are the same." }}}}{MARK "3 0 0" 0 }{VIEWOPTS 1 1 0 3 2 
1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }