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1
THE CHAIN RULE
The whole purpose of the chain rule is to be able to find the derivative of a complicated function
without to much stress. Consider the following composition of functions:
p
(x) =
f
(
g
(
k
(
h
(
x
)))). The
derivative
p
' (x) =
f
' (
g
(
k
(
h
(
x
))))*
g
' (
k
(
h
(
x
)))*
k
' (
h
(
x
))*
h
' (
x
). Determining this derivative is
like peeling an onion from the outside in. Take the derivative of the outer function, times the derivative
of the next layer. Keep doing this until you get to the final innermost layer.
The Chain Rule is formally stated and proved at the end of these notes.
EXAMPLE 1:
Find
f
' for
f
(
x
) = (
x
4
+ 2
x
2
 6)
100
.
SOLUTION:
Let
u
=
x
4
2
x
2
−
6 and
f
u
=
u
100
.
Then
u'
=
4
x
3
4
x
f '
u
=
100
u
99
f '
x
= f '
u
x
⋅
u'
x
=
100
u
100
4
x
3
4
x
=
100
x
4
2
x
2
−
6
99
4
x
3
4
x
Remember the rule for
f
(
x
) =
x
n
for which
f
' (
x
) =
n
x
n
– 1
? n can be any real number, even a
fraction.
Notice, that the final answer in Example 1 is the derivative of the inside times the derivative of the
outside. Using the chain rule is like peeling an onion. Start with the outermost function and take its
derivative, then the next function, and so on.
The next example illustrates the Chain Rule in combination with the product rule. It is very
important that you gain facility in applying multiple rules in combination to the point that it becomes
second nature to you!
EXAMPLE 2:
Find
f '
for
f
x
=
1
x
−
2
x
5
3
=
x
−
2
x
5
 3
SOLUTION:
Use the product rule to find
u
'. (
f
(
x
) =
h
(
x
) *
g
(x).
f
' (
x
) =
h
' (
x
)*
g
(
x
) +
h
(
x
)*
g
' (
x
).)
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EXAMPLE 3:
Find
f '
for
f
x
=
3
x
2
3
x
4.
SOLUTION:
Note that in this problem we have the function
x
2
3
x
4
raised to the
1
3
power.
This is thus
a composite function which requires the Chain Rule.
In differentiating the outer function we have to
use the “drop down” rule for the outer function.
We begin by letting
u
represent the the inner function.
This is a procedure you should write out each time you do this type of problem, at least for a period of
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 Spring '08
 Noohi
 Chain Rule, Derivative, The Chain Rule

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