11DIFFERENTIATION - Department of Mathematics

722 11 DIFFERENTIATION
FIGURE 11.5
The composite function h(x) g[ f(x)]
Rule 7: The Chain Rule
the function f, we might suspect that the rule h(x) for the derivative h of h
will be given by an expression that involves the rules for the derivatives of f
and g. But how do we combine these derivatives to yield h?
This question can be answered by interpreting the derivative of each
function as giving the rate of change of that function. For example, suppose
u f(x) changes three times as fast as x—that is,
f(x) du
3
dx
And suppose y g(u) changes twice as fast as u—that is,
g(u) dy
2
du
Then, we would expect y h(x) to change six times as fast as x—that is,
h(x) g(u) f(x) (2)(3) 6
or equivalently,
dy dy
dx du
du
(2)(3) 6
dx
This observation suggests the following result, which we state without proof.
If h(x) g[ f(x)], then
h(x) d
g( f(x)) g( f(x)) f(x) (1)
dx
Equivalently, if we write y h(x) g(u), where u f(x), then
REMARKS
h = g[ f ]
f g
x f (x) = u y = g(u) = g[ f (x)]
dy dy du
dx du dx
1. If we label the composite function h in the following manner
Inside function
h(x) g[ f(x)]
Outside function
then h(x) is just the derivative of the ‘‘outside function’’ evaluated at the
‘‘inside function’’ times the derivative of the ‘‘inside function.’’
(2)