Page 91 - Calculus Workbook For Dummies

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

Linking Up with the Chain Rule

The chain rule is probably the trickiest among the advanced rules, but it’s really not
that bad at all if you focus clearly on what’s going on — I promise! Most of the basic
derivative rules have a plain old x as the argument (or input variable) of the function.
x
For example, f x = x, sin x, and y = e all have just x as the argument.
^ h
When the argument of a function is anything other than a plain old x, such as y = sinx 2
x
or ln10 , you’ve got a chain rule problem.
Here’s what you do. You simply apply the derivative rule that’s appropriate to the
outer function, temporarily ignoring the not-a-plain-old-x argument. Then multiply that
result by the derivative of the argument. That’s all there is to it.

Q. What’s the derivative of y = sinx ? Q. What’s the derivative of sin x ? You’ve got
3
3
4
to use the chain rule twice for this one.
A. y = l 3 x 2 cos x 3
A. The answer is 12 x 2 sin x $ cos x 3
3
3
3
1. Temporarily think of the argument, x ,
4
3
as a glob. 1. Rewrite sin x to show what it really
4
3
means: sin x i
_
So, you’ve got y = sin(glob).
2. The outermost function is the 4th power,
2. Use the regular derivative rule.
so use the derivative rule for stuff —
4
y = sin globi , so that’s stuff4 3 ; then multiply that by the
_
3
derivative of the inside stuff, sin x .
y = l cos globi 3
_
3 l
3
4 _ sinx $ _i sinx i
(This is only a provisional answer, so the
“=” sign is false — egad! The math police With chain rule problems, always work
are going to pull me over.) from the outside, in.
3
3. Multiply this by the derivative of the 3. To get the derivative of sin x , use the
argument. derivative rule for sin globi, and then
_
multiply that by globl.
y = l cos glob $ i globl
_
3 3 l
3
3
i
4 _ sinx $ cos x $ _i x i
_
4. Get rid of the glob.
3
2
4. The derivative of x is 3x , so you’ve got
3
2
The glob equals x so globl equals 3x . 3
3
3
_
i
4 _ sinx $ cos x $ i 3 x 2
3
y = l cos x $ i 3 x 2
_
5. To simplify, rewrite the sin power and
= 3 x 2 cosx 3 2
move the 3x to the front.
3
3
= 12 x 2 sin x $ cosx 3
Can you remember when you’ve had so
much fun?
With chain rule problems, never use
more than one derivative rule per step. In
other words, when you do the derivative
rule for the outermost function, don’t
touch the inside stuff! Only in the next
step do you multiply the outside deriva-
tive by the derivative of the inside stuff.