Page 101 - Calculus Workbook For Dummies

85
Chapter 6: Rules, Rules, Rules: The Differentiation Handbook

x
x
*p d sinx = x cos lnx - 3 sin lnx - sinx
dx x 3 lnx x ^ lnxh 2
4
i
h
d sinx ^ sinx l _ x 3 lnx - ^ sinx _ h x 3 lnxi l
dx x 3 lnx = _ x 3 lnxi 2
Product Rule
64444444 74444444 8
3 l
3
i
^ cosx _ h x 3 lnx - ^ sinx a h _ x i ^ lnx + _ x ^ i lnx l
h
h k
=
6 2
x ^ lnxh
1
3
x 3 cos lnx - ^ sinx e h 3 x 2 lnx + _ x c i x mo
x
=
6 2
x ^ lnxh
x 3 cos lnx - 3 x 2 sin lnx - x 2 sinx
x
x
= 2
6
x ^ lnxh
x cos lnx - 3 sin lnx - sinx
x
x
=
4 2
x ^ lnxh
q f x = sinx 2 ; f l ^ h 2 x cosx 2
x =
^ h
Because the argument of the sine function is something other than a plain old x, this is a chain
2
rule problem. Just use the rule for the derivative of sine, not touching the inside stuff (x ), then
2
multiply your result by the derivative of x .
2
x =
_
f l ^ h cos x $ i 2 x
= 2 x cosx 2
r g x = sin x ; g x = 3sin 2 xcos x
3
l ^ h
^ h
3
3
Rewrite sin x as sinxh so that it’s clear that the outermost function is the cubing function.
^
2
3
3
By the chain rule, the derivative of stuff is stuff $ stuff'. The stuff here is sin x and thus stuff' is
2 2
cos x. So your final answer is 3^ sinx $ h cosx, or sin x3 cosx.
s s t = tan lnth ; s t = sec lnt $ h 1 t
2
^
l ^ h
^
^ h
2 2
The derivative of tan is sec , so the derivative of tan(lump) is sec lump $ i lumpl. You better
_
1 1
2
^
know by now that the derivative of ln t is , so your final result is sec lnt $ h .
t t
t y = e 4 x 3 ; y = l 12 x e 4 x 3
2
x
x
glob
The derivative of e is e , so by the chain rule, the derivative of e glob is e $ globl.
3
2
4
x
So y = l e $ 12 x , or x e12 2 4 x 3 .
*u f x = x 4 sin x ; f l ^ h 4 x 3 sin x + 3 x 4 sin x cosx
3
2
3
x =
^ h
This problem involves both the product rule and the chain rule. Which do you do first? Note
3
that the chain rule part of this problem, sin x, is one of the two things being multiplied, so it is
part of — or sort of inside — the product. And, like with pure chain rule problems, with prob-
lems involving more than one rule, you work from outside, in. So here you begin with the prod-
uct rule. Here’s another way to look at it:
If you’re not sure about the order of the rules in a complicated derivative problem, imagine that
you plugged a number into x in the original function and had to compute the answer. Your last
3
computation tells you where to start. If, for example, you plugged 2 into x 4 sin x, you would
3
4
4
3
compute 2 , then sin 2, then you’d cube that to get sin 2, and, finally, you’d multiply 2 by sin 2.
Because your final step was multiplication, you begin with the product rule.