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

206 Chapter 2 • The Derivative and Its Properties

2 Differentiate the Quotient of Two Functions

The derivative of the quotient of two functions is not equal to the quotient of their
derivatives. Instead, the derivative of the quotient of two functions is found using the
Quotient Rule.

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THEOREM Quotient Rule
f (x)
If two functions f and g are differentiable and if F(x) = , g(x) 6= 0, then F
g(x)
is differentiable, and the derivative of the quotient F is
′ ′ ′
f (x) f (x)g(x) − f (x)g (x)
′
F (x) = = 2
g(x) [g(x)]
IN WORDS The derivative of a quotient of
two functions is the derivative of the
In Leibniz notation, the Quotient Rule has the form
numerator times the denominator, minus the
numerator times the derivative of the

denominator, all divided by the denominator d f (x) g(x) − f (x) d g(x)
squared. That is, d d f (x) dx dx
F(x) = =
′ 2
′
f f g − f g ′ dx dx g(x) [g(x)]
=
g g 2
Proof We use the definition of a derivative (Form 2) to find F (x).
′
f (x + h) f (x)
−
F(x + h) − F(x) g(x + h) g(x)
F (x) = lim = lim
′
h→0 h ↑ h→0 h
f (x)
F(x) =
g(x)
f (x + h)g(x) − f (x)g(x + h)
= lim
h→0 h[g(x + h)g(x)]
We write F in an equivalent form that contains the difference quotients for f and g
′
by subtracting and adding f (x)g(x) to the numerator.
f (x + h)g(x) − f (x)g(x) + f (x)g(x) − f (x)g(x + h)
F (x) = lim
′
h→0 h[g(x + h)g(x)]
Now group and factor the numerator.
[ f (x + h) − f (x)]g(x) − f (x)[g(x + h) − g(x)]
F (x) = lim
′
h→0 h[g(x + h)g(x)]

f (x + h) − f (x) g(x + h) − g(x)
g(x) − f (x)
h h
= lim
h→0 g(x + h)g(x)

f (x + h) − f (x) g(x + h) − g(x)
lim · lim g(x) − lim f (x) · lim
h→0 h h→0 h→0 h→0 h
=
lim g(x + h) · lim g(x)
h→0 h→0
RECALL Since g is differentiable, it is f (x)g(x) − f (x)g (x)
′
′
continuous; so, lim g(x + h) = g(x). = 2
h→0 [g(x)]

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