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Proof:
f 1g1x h22 f 1g1x22
By the definition of the derivative, 3 f 1g1x22 4 ¿ lim
.
hS0
h
Assuming that g1x h2 g1x2 0, we can write
f 1g1x h22 f 1g1x22 g1x h2 g1x2
3 f 1g1x22 4 ¿ lim ca ba bd
hS0 g1x h2 g1x2
h
f 1g1x h22 f 1g1x22
lim c
hS0 g1x h2 g1x2
Therefore, 3 f 1g1x22 4 ¿ f ¿1g1x22g¿1x2.
g1x h2 g1x2
d lim c
d
hS0 h
Since lim 3g1x h2 g1x24 0, let g1x h2 g1x2 k and k S 0
hS0
as h S 0. We obtain
f 1g1x2 k2 f 1g1x22 h2 g1x2
3 f 1g1x22 4 ¿ lim c d lim cg1x
kS0 k
hS0 h
(Property of
limits)
This proof is not valid for all circumstances. When dividing by g1x h2 g1x2,
we assume that g1x h2 g1x2 0. A proof that covers all cases can be found
in advanced calculus textbooks.
d
EXAMPLE 2
Applying the chain rule
Differentiate h1x2 1x 2 x2 3 2 .
Solution
The inner function is
g1x2 x 2 x, and the outer function is f 1x2 x 3 2 .
The derivative of the inner function is g¿1x2 2x 1.
The derivative of the outer function is f ¿1x2 3 2 x1 2 .
The derivative of the outer function evaluated at the inner function g1x2
f ¿1x 2 x2 3 2 1x2 x2 1 2 .
is
By the chain rule, h¿1x2 3 2 1x2 x2 1 2 12x 12.
The Chain Rule in Leibniz Notation
If y is a function of u and u is a function of x (so that y is a composite
dy
dy du
function), then provided that and exist.
dx dy du
du dx ,
du dx
If we interpret derivatives as rates of change, the chain rule states that if y is a
function of x through the intermediate variable u, then the rate of change of y
100
2.5 THE DERIVATIVES OF COMPOSITE FUNCTIONS
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