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2.3. THE PRODUCT AND QUOTIENT RULES
Summary
In this section, we encountered the following important ideas:
• If a function is a sum, product, or quotient of simpler functions, then we can use the
sum, product, or quotient rules to differentiate the overall function in terms of the
simpler functions and their derivatives.
• The product rule tells us that if P is a product of differentiable functions f and g
according to the rule P(x) = f (x)g(x), then
P
′ (x) = f (x)g
′ (x) + g(x) f
′ (x).
• The quotient rule tells us that if Q is a quotient of differentiable functions f and g
according to the rule Q(x) =
f (x)
g(x) , then
Q
′ (x) =
g(x) f ′ (x) − f (x)g ′ (x)
g(x) 2
.
• The product and quotient rules now complement the constant multiple and sum
rules and enable us to compute the derivative of any function that consists of sums,
constant multiples, products, and quotients of basic functions we already know how to
differentiate. For instance, if F has the form
F(x) =
2a(x) − 5b(x)
c(x) · d(x)
,
then F is fundamentally a quotient, and the numerator is a sum of constant multiples
and the denominator is a product. Hence the derivative of F can be found by applying
the quotient rule and then using the sum and constant multiple rules to differentiate
the numerator and the product rule to differentiate the denominator.
Exercises
1. Let f and g be differentiable functions for which the following information is known:
f (2) = 5, g(2) = −3, f ′ (2) = −1/2, g ′ (2) = 2.
(a) Let h be the new function defined by the rule h(x) = g(x) · f (x). Determine
h(2) and h ′ (2).
(b) Find an equation for the tangent line to y = h(x) at the point (2, h(2)) (where h
is the function defined in (a)).
(c) Let r be the function defined by the rule r(x) =
g(x)
f (x) . Is r increasing, decreasing,
or neither at a = 2? Why?
2.3. THE PRODUCT AND QUOTIENT RULES
Summary
In this section, we encountered the following important ideas:
• If a function is a sum, product, or quotient of simpler functions, then we can use the
sum, product, or quotient rules to differentiate the overall function in terms of the
simpler functions and their derivatives.
• The product rule tells us that if P is a product of differentiable functions f and g
according to the rule P(x) = f (x)g(x), then
P
′ (x) = f (x)g
′ (x) + g(x) f
′ (x).
• The quotient rule tells us that if Q is a quotient of differentiable functions f and g
according to the rule Q(x) =
f (x)
g(x) , then
Q
′ (x) =
g(x) f ′ (x) − f (x)g ′ (x)
g(x) 2
.
• The product and quotient rules now complement the constant multiple and sum
rules and enable us to compute the derivative of any function that consists of sums,
constant multiples, products, and quotients of basic functions we already know how to
differentiate. For instance, if F has the form
F(x) =
2a(x) − 5b(x)
c(x) · d(x)
,
then F is fundamentally a quotient, and the numerator is a sum of constant multiples
and the denominator is a product. Hence the derivative of F can be found by applying
the quotient rule and then using the sum and constant multiple rules to differentiate
the numerator and the product rule to differentiate the denominator.
Exercises
1. Let f and g be differentiable functions for which the following information is known:
f (2) = 5, g(2) = −3, f ′ (2) = −1/2, g ′ (2) = 2.
(a) Let h be the new function defined by the rule h(x) = g(x) · f (x). Determine
h(2) and h ′ (2).
(b) Find an equation for the tangent line to y = h(x) at the point (2, h(2)) (where h
is the function defined in (a)).
(c) Let r be the function defined by the rule r(x) =
g(x)
f (x) . Is r increasing, decreasing,
or neither at a = 2? Why?