Page 102 - Calculus Workbook For Dummies

86 Part III: Differentiation

3
f x = x 4 sin x
^ h
4 3
= _ x _i sin xi
4 l 3 4 3 l
f l ^ h x i _ sin x + _i x _i sin xi _ product rulei
x = _
3 l 3 3
_
Use the chain rule to solve sin xi , then go back and finish the problem. sin x means sinxh
^
3
3
2
3
and that’s stuff . The derivative of stuff is stuff $ stuff', so the derivative of sinxh is
3
^
2
3^ sinx $ h cosx. Now continue the solution.
4 l 3 4 3
x = _
f l ^ h x i _ sin x + _i x _i sin xi l
3
4
3
= 4 x 3 sin x + x $ ^ sinxh 2 cosx
3
2
= 4 x 3 sin x + 3 x 4 sin x cosx
2 2
^ lnxh 2 lnx 5 ^ lnxh
*v g x = ; g x = -
l ^ h
^ h
x
5 x - 4 x 5 - 4h ^ 5 x - 4h 2
^
Here you’ve got the chain rule inside the quotient rule. Start with the quotient rule:
2 l 2
4 l
^ a lnxh k 5 ^ x - 4 - ^h lnx ^h 5 x - h
g x = 2
l ^ h
5 ^ x - 4h
2 l 2
Next, take care of the chain rule solution for lnxh k . You want the derivative of glob — that’s
^ a
2 1
$
2 glob globl. So the derivative of lnxh is 2^ lnx c h x m. Now you can finish:
^
1 2
2^ lnx c h x 5 ^ m x - 4 - ^h lnx ^h 5h
g x = 2
l ^ h
5 ^ x - 4h
2
^ 10 x - 8h lnx - 5 x ^ lnxh
= 2
x 5 - 4h
x
^
2
2 lnx 5^ lnxh
= -
x
x 5 - 4h 5 ^ x - 4h 2
^
*w y = cos x 2 ; y = - 24 x cos 4 x 2 sin x 2
2
3
4
l
4
Triply nested!
3
2
4
y = _ cos x i
3
2
3
The derivative of stuff is stuff $ stuff', so, you’ve got,
2 2 l
2
4
4
y = l 3 _ cos x $ _i cos x i
-
_
Now you do the derivative of cos globi, which is sin glob $ i globl. Two down, one to go:
_
2 2 l
y = l 3 _ cos x i _ - sin x $ _i 4 x i
2
2
4
4
2
2
2
4
4
= 3 cos x - sin x $ i 8 x
_
2
= - 24 x cos x 2 sin x 2
4
4