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Sullivan 04 apcalc4e 45342 ch02 166 233 5pp August 7, 2023 12:54

Section 2.4 • Differentiating the Product and the Quotient of Two Functions; Higher-Order Derivatives 203

Retain Your Knowledge
Multiple-Choice Questions
x 2
1. The graph of a piecewise defined function f is shown in the 2. lim ln x =
accompanying figure. Use the graph to determine which of the x → 1 e
following statements are true. 1
(A) −1 (B) (C) 1 (D) 2
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I. lim f (x) = 1 e
x→2 − ( x 2 if x < 2
II. lim f (x) does not exist
3. Given f (x) = 2 if x = 2 ,
x→2 +
III. lim f (x) = f (4) x + 1 if x > 2
x→4
find lim f (x), if it exists.
y x→2
(22, 4)
4 y  f(x) (A) 2 (B) 3 (C) 4 (D) The limit does not exist.
Free-Response Question
2
(2, 1) 1
3
4. Show that lim x sin = 0.
22 2 4 x x→0 x
(Hint: Use the Squeeze Theorem.)
22
(A) I only (B) I and II only
(C) I and III only (D) I, II, and III

2.4 Differentiating the Product

and the Quotient of Two Functions;
Higher-Order Derivatives

OBJECTIVES When you finish this section, you should be able to:
1 Differentiate the product of two functions (p. 203)
2 Differentiate the quotient of two functions (p. 206)
3 Find higher-order derivatives (p. 208)
4 Find the acceleration of an object moving on a line (p. 210)

In this section, we obtain formulas for differentiating products and quotients of
functions. As it turns out, the formulas are not what we might expect. The derivative
of the product of two functions is not the product of their derivatives, and the derivative
of the quotient of two functions is not the quotient of their derivatives.

1 Differentiate the Product of Two Functions

3
Consider the two functions f (x) = 2x and g(x) = x . Both are differentiable, and their
2
derivatives are f (x) = 2 and g (x) = 3x . Form the product
′
′
3
F(x) = f (x)g(x) = 2x · x = 2x 4
Now find F using the Constant Multiple Rule and the Simple Power Rule.
′
3
′
F (x) = 2 · 4x = 8x 3
d
′ ′ 2 2 ′ 3
Notice that f (x)g (x) = 2 · 3x = 6x is not equal to F (x) = [ f (x)g(x)] = 8x . We
dx
conclude that the derivative of a product of two functions is not the product of their
derivatives.
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