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

204 Chapter 2 • The Derivative and Its Properties

To find the derivative of the product of two differentiable functions f and g, we
let F(x) = f (x)g(x) and use the definition of a derivative, namely,

[ f (x + h)g(x + h)] − [ f (x)g(x)]
F (x) = lim Form (2)
′
h→0 h
We can express F in an equivalent form that contains the difference quotients for f
′
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and g, by subtracting and adding f (x + h)g(x) to the numerator.
f (x + h)g(x + h) − f (x + h)g(x) + f (x + h)g(x) − f (x)g(x)
F (x) = lim
′
h→0 h
f (x + h)[g(x + h) − g(x)] + [ f (x + h) − f (x)]g(x)
= lim Group and factor.
h→0 h

h i g(x + h) − g(x) f (x + h) − f (x) h i
= lim f (x + h) lim + lim lim g(x) Use properties of limits.
h→0 h→0 h h→0 h h→0
h i h i
′
′
= lim f (x + h) g (x) + f (x) lim g(x) Definition of a derivative.
h→0 h→0
= f (x)g (x) + f (x)g(x) lim g(x) = g(x) since h is not present.
′
′
h→0
Since f is differentiable, it is
continuous, so lim f (x + h) = f (x).
h→0
We have proved the following theorem.

THEOREM Product Rule
If f and g are differentiable functions and if F(x) = f (x)g(x), then F is
differentiable, and the derivative of the product F is

′
′
F (x) = [ f (x)g(x)] = f (x)g (x) + f (x)g(x)
′
′
IN WORDS The derivative of the product of
In Leibniz notation, the Product Rule has the form
two differentiable functions equals the first
function times the derivative of the second
function plus the derivative of the first d d d d
function times the second function. That is, F(x) = [ f (x)g(x)] = f (x) g(x) + f (x) g(x)
dx dx dx dx
′
( f g) = f · g + f · g
′
′
EXAMPLE 1 Differentiating the Product of Two Functions

2
x
Find y if y = (1 + x )e .
′
Solution
2
The function y is the product of two functions: a polynomial, f (x) = 1 + x , and the
x
exponential function, g(x) = e . By the Product Rule,
d d d
′ 2 x 2 x 2 x 2 x x
y = [(1 + x )e ] = (1 + x ) e + (1 + x ) e = (1 + x )e + 2xe
dx ↑ dx dx
Product Rule

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