12.3 The Chain and General Power Rules
7) The Chain Rule
If
[
]
)
(
)
(
x
f
g
x
h
=
, then:
′
=
=
′
⋅
′
h
x
d
dx
g f
x
g
f
x
f
x
( )
(
( ))
(
( ))
( )
Equivalently, if we let
)
(
x
f
U
=
we would have
)
(
)
(
U
g
x
h
=
and thus
)
(
)
(
U
g
x
h
y
=
=
, then:
dy
dx
dy
dU
dU
dx
=
⋅
Many composite functions have the form
[
]
n
x
f
x
h
)
(
)
(
=
where n is a real number.
In these cases a shortcut
formula called the general power rule can be used to find
′
h
x
( )
.
The General Power Rule
If the function
f
is differentiable and
[
]
n
x
f
x
h
)
(
)
(
=
(
n
is a real number), then:
′
=
=
⋅
′
−
h
x
d
dx
f
x
n f
x
f
x
n
n
( )
[
( )]
[
( )]
( )
1
Example 1
Find
′

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