11DIFFERENTIATION - Department of Mathematics

11.3 The Chain Rule
11.3 THE CHAIN RULE 721
This section introduces another rule of differentiation called the chain rule.
When used in conjunction with the rules of differentiation developed in the
last two sections, the chain rule enables us to greatly enlarge the class of
functions we are able to differentiate.
T HE C HAIN R ULE
Consider the function h(x) (x2 x 1) 2 . If we were to compute h(x)
using only the rules of differentiation from the previous sections, then our
approach might be to expand h(x). Thus,
h(x) (x2 x 1) 2 (x2 x 1)(x2 x 1)
x4 2x3 3x2 2x 1
from which we find
h(x) 4x3 6x2 6x 2
But what about the function H(x) (x2 x 1) 100 ? The same technique
may be used to find the derivative of the function H, but the amount of work
involved in this case would be prodigious! Consider, also, the function
G(x) x2 1. For each of the two functions H and G, the rules of differentiation
of the previous sections cannot be applied directly to compute the
derivatives H and G.
Observe that both H and G are composite functions; that is, each is
composed of, or built up from, simpler functions. For example, the function
H is composed of the two simpler functions f(x) x2 x 1 and g(x) x100 as follows:
H(x) g[ f(x)] [ f(x)] 100
(x 2 x 1) 100
In a similar manner, we see that the function G is composed of the two simpler
functions f(x) x2 1 and g(x) x. Thus,
G(x) g[ f(x)] f(x)
x2 1
As a first step toward finding the derivative h of a composite function
h g f defined by h(x) g[ f(x)], we write
u f(x) and y g[ f(x)] g(u)
The dependency of h on g and f is illustrated in Figure 11.5. Since u is a
function of x, we may compute the derivative of u with respect to x, iff is a
differentiable function, obtaining du/dx f(x). Next, if g is a differentiable
function of u, we may compute the derivative of g with respect to u, obtaining
dy/du g(u). Now, since the function h is composed of the function g and