1.7 The Chain Rule

Math 135 Class Notes

1.7

Business Calculus

Spring 2009

The Chain Rule

THE EXTENDED POWER RULE According to the Power Rule, the derivative of the power function y = xk is given by d k (x ) = kxk−1 . dx 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 §k d £ g(x) dx £ §2 In particular, suppose we want the derivative of y = g(x) . We can compute this using the Product Rule: §2 § d £ d £ g(x) = g(x) · g(x) dx dx = g 0 (x) · g(x) + g(x) · g 0 (x) = 2g(x) · g 0 (x) This resembles the result of differentiating y = x2 . The exponent 2 has been pulled down in front of the function and the power has been reduced by 1. However, we also have a factor of g 0 (x) that appears in the derivative. This is a special case of the following general rule: THEOREM 7 The Extended Power Rule Suppose that g(x) is a differentiable function of x. Then, for any real number k, §k £ §k−1 d £ § £ §k−1 0 d £ g(x) = k g(x) · g(x) = k g(x) · g (x). dx dx

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. derivative of power function

z }| { §k £ §k−1 d £ g(x) = k g(x) · dx EXAMPLE

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

27

g 0 (x) | {z }

derivative of inside function

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Chapter 1

EXAMPLE

Differentiation

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

r 4

x+3 . x−2

EXAMPLE

Differentiate f (x) =

EXAMPLE

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

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The Chain Rule

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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 to compute 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 the house. 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 A . 250 Suppose a gallon of paint costs $27.50. Then the cost C of n gallons of paint is given by the function n = g(A) =

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 itself a function of the area A to be painted, n = g(A). The cost to paint an area A is then a “function of a function,” 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 and g by two machines, we can visualize the composition of functions as shown in the figure. The input to g is x. The output g(x) of g is used as the to f . The ° input ¢ output to f is then f g(x) .

30

Chapter 1

EXAMPLE

Differentiation

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

1 . Find x

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.” outside function z ° }| ¢{ (f ◦ g)(x) = f g(x) |{z} inside function

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

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

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The Chain Rule

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The next theorem now tells us how to differentiate a composition of functions. THEOREM 8 The Chain Rule The derivative of the composition f ◦ g is given by § ¢§ ° ¢ d £ d £ ° (f ◦ g)(x) = f g(x) = f 0 g(x) · g 0 (x) = dx dx In Leibniz’s notation, if y = f (u) and u = g(x), then

derivative of outer function

z ° }| ¢{ f 0 g(x) ·

g 0 (x) | {z }

derivative of inner function

dy dy du = · . dx du dx This says that the derivative of the composition is the product of the derivatives of the outside and inside 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.