J. Kim MS 125 Worksheet 3.4 (Sep. 29, 2006)
Section 3.4 The Chain Rule
Example 1 Express each function as a composition of two functions.
1.
5
2
34)( xxF
2.
13
)(
x
exG
3.
253)(
2
xxxH
4.
4ln)(
2
xxT
The Chain Rule
If
)(xf
and
)(xg
are differentiable functions, then the composition function
)(xgf
the two
functions is differentiable, and the derivative of is
)()()( xgxgfxgf
dx
d
In words, the derivative of a composite function is the product of the derivatives of the outside and
inside functions. The derivative of the outside function must be evaluated at the inside function.
Example 2 Find the derivatives of the following functions by using the chain rule.
1.
102
)1()( xxf
2.
253)(
2
xxxg
3.
24
1
)( xx
xh
4.
35
)(
x
exf
5.
2
)(
x
exg
6.
5)(
2
x
exh
7.
19
2
1)(
t
exf
Example 3 Find the derivatives of the following functions by using appropriate rules and formulas.
1.
143
)(
x
exxf
2.
1
1
)(
3
x
x
e
e
xh