1.7 the chain rule

Math 135 Business Calculus Spring 2009Class Notes1.7 The Chain Rule

THE EXTENDED POWER RULE

According to the Power Rule, the derivative of the power function y = xk is given byd

dx(xk) = kxk−1.

For a power function y = xk, we have a variable x raised to a power k. What is the derivative if,instead of the variable x raised to a power, we have a function raised to a power, such as

d

dx

£g(x)

§k

In particular, suppose we want the derivative of y =£g(x)

§2. We can compute this using the ProductRule:

d

dx

£g(x)

§2 =d

dx

£g(x) · g(x)

§

= g0(x) · g(x) + g(x) · g0(x)= 2g(x) · g0(x)

This resembles the result of differentiating y = x2. The exponent 2 has been pulled down in front ofthe function and the power has been reduced by 1. However, we also have a factor of g0(x) that appearsin the derivative. This is a special case of the following general rule:

THEOREM 7 The Extended Power RuleSuppose that g(x) is a differentiable function of x. Then, for any real number k,

d

dx

£g(x)

§k = k£g(x)

§k−1 · d

dx

£g(x)

§= k

£g(x)

§k−1 · g0(x).

In this rule, we can think of g(x) as the “inside” function and the kth power as the “outside”function. The rule can then be viewed as follows.

d

dx

£g(x)

§k =

derivative ofpower functionz }| {k£g(x)

§k−1 · g0(x)| {z }

derivative ofinside function

EXAMPLE Differentiate f(x) = (7x2 + x3)5.

27

28 Chapter 1 Differentiation

EXAMPLE Differentiate h(x) = (7x2 + x3)1/2.

EXAMPLE Differentiate f(x) = 4

rx + 3x− 2

.

EXAMPLE Differentiate f(x) = (3x− 5)4(7− x)10.

1.7 The Chain Rule 29

COMPOSITION OF FUNCTIONS AND THE CHAIN RULE

The Extended Power Rule is a special case of a more general differentiation rule that can be used tocompute the derivative of the composition of any two functions.

EXAMPLE The number of gallons of paint needed to paint a house depends upon the size of thehouse. A gallon of paint typically covers 250 square feet. This means the number of gallons of paint,n, is a function of the area, A, to be painted according to the function

n = g(A) =A

250.

Suppose a gallon of paint costs $27.50. Then the cost C of n gallons of paint is given by the function

C = f(n) = 27.5n.

Find a function for the cost C to paint an area of A square feet.

In the above example, the cost C of n gallons of paint is a function of n, C = f(n), where n is itselfa function of the area A to be painted, n = g(A). The cost to paint an area A is then a “function of afunction,” or a composite function. If the function giving C in terms of A is denoted h, so C = h(A),then we write

C = h(A) = (f ◦ g)(A).The function h is called the composition of the functions f and g.

DEFINITION OF COMPOSITION OF FUNCTIONS

Suppose f(x) and g(x) are two functions. The composition of f and g, denoted f ◦ g, is defined by

(f ◦ g)(x) = f°g(x)

¢.

If we represent the two functions f andg by two machines, we can visualize thecomposition of functions as shown in thefigure. The input to g is x. The outputg(x) of g is used as the input to f . Theoutput to f is then f

°g(x)

¢.

30 Chapter 1 Differentiation

EXAMPLE Let f(x) = x3 and g(x) =1x

. Find

a) (f ◦ g)(x)

b) (g ◦ f)(x)

Given a composition (f ◦ g)(x) = f°g(x)

¢, we can think of g as the “inside function” and f as the

“outside function.”

(f ◦ g)(x) =

outside functionz }| {f°

g(x)|{z}inside

function

¢

We sometimes have to “decompose” a complicated function and write it as the composition of twosimpler functions.

EXAMPLE Suppose h(x) =1√

7x + 2. Find functions f(x) and g(x) such that h(x) = (f ◦ g)(x).

1.7 The Chain Rule 31

The next theorem now tells us how to differentiate a composition of functions.

THEOREM 8 The Chain RuleThe derivative of the composition f ◦ g is given by

d

dx

£(f ◦ g)(x)

§=

d

dx

£f°g(x)

¢§= f 0

°g(x)

¢· g0(x) =

derivative ofouter functionz }| {

f 0°g(x)

¢· g0(x)

| {z }derivative ofinner function

In Leibniz’s notation, if y = f(u) and u = g(x), thendy

dx=

dy

du· du

dx.

This says that the derivative of the composition is the product of the derivatives of the outside andinside functions.

EXAMPLE Suppose f(u) = 2 +√

u and u = g(x) = x3 + 1. Find (f ◦ g)0(x)a) By using the Chain Rule.

b) By first finding (f ◦ g)(x) and then differentiating.