10. The Chain Rule
P. K. Lamm
10/02/11 (14:44)
p. 1 / 5
Lecture Notes:
10. The Chain Rule
These classnotes are intended to be supplementary to the textbook and are necessarily limited
by the time allotted for classes. For full and precise statements of deﬁnitions and theorems,
as well as material covering other topics and examples, please consult the textbook.
1. The Chain Rule
We have seen some rules for derivatives such as
(
x
n
)
0
=
nx
n

1
,
for integer
n
6
= 0. The situation changes however when, instead of raising
x
to a power, we raise
a
function of
x
to a power. For example, for
g
diﬀerentiable we have from the product rule that
(
g
2
(
x
))
0
= (
g
(
x
)
·
g
(
x
))
0
=
g
(
x
)
g
0
(
x
) +
g
0
(
x
)
g
(
x
)
.
Thus
(
g
2
(
x
))
0
= 2
g
(
x
)
·
g
0
(
x
)
instead of 2
g
(
x
) as we might have expected. Similarly, we may use the formula for the derivative
of a product of three factors to obtain
(
g
3
(
x
))
0
= (
g
(
x
)
g
(
x
)
g
(
x
))
0
=
g
0
(
x
)
g
(
x
)
g
(
x
) +
g
(
x
)
g
0
(
x
)
g
(
x
) +
g
(
x
)
g
(
x
)
g
0
(
x
)
or that
(
g
3
(
x
))
0
= 3
g
2
(
x
)
g
0
(
x
)
.
In fact it’s true that for integer
n
6
= 0,
(
g
n
(
x
))
0
=
ng
n

1
(
x
)
g
0
(
x
)
.
(1)
Let’s introduce some notation. Let
f
(
u
) =
u
n
,
and
u
(
x
) =
g
(
x
)
.
Then with this notation we may write
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 Fall '10
 KIHYUNHYUN
 Calculus, Chain Rule, Derivative, The Chain Rule, P. K. Lamm

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