2.3. THE PRODUCT AND QUOTIENT RULES
103
asks you to investigate the derivative of a product and quotient of two polynomials.
Preview Activity 2.3. Let f and g be the functions defined by f (t) = 2t 2 and g(t) = t 3 +4t.
(a) Determine f ′ (t) and g ′ (t).
(b) Let p(t) = 2t 2 (t 3 + 4t) and observe that p(t) = f (t) · g(t). Rewrite the formula for
p by distributing the 2t 2 term. Then, compute p ′ (t) using the sum and constant
multiple rules.
(c) True or false: p ′ (t) = f ′ (t) · g ′ (t).
(d) Let q(t) =
t 3 + 4t
2t 2 and observe that q(t) =
g(t)
f (t)
. Rewrite the formula for q by
dividing each term in the numerator by the denominator and simplify to write q
as a sum of constant multiples of powers of t. Then, compute q ′ (t) using the sum
and constant multiple rules.
(e) True or false: q
′ (t) =
g ′ (t)
f ′ (t)
.
⊲⊳
The product rule
As parts (b) and (d) of Preview Activity 2.3 show, it is not true in general that the derivative
of a product of two functions is the product of the derivatives of those functions. Indeed, the
rule for differentiating a function of the form p(x) = f (x) · g(x) in terms of the derivatives
of f and g is more complicated than simply taking the product of the derivatives of f and
g. To see further why this is the case, as well as to begin to understand how the product
rule actually works, we consider an example involving meaningful functions.
Say that an investor is regularly purchasing stock in a particular company. Let N(t) be
a function that represents the number of shares owned on day t, where t = 0 represents
the first day on which shares were purchased. Further, let S(t) be a function that gives the
value of one share of the stock on day t; note that the units on S(t) are dollars per share.
Moreover, to compute the total value on day t of the stock held by the investor, we use the
function V (t) = N(t) · S(t). By taking the product
V (t) = N(t) shares · S(t) dollars per share,
we have the total value in dollars of the shares held. Observe that over time, both the
number of shares and the value of a given share will vary. The derivative N ′ (t) measures
the rate at which the number of shares held is changing, while S ′ (t) measures the rate at
103
asks you to investigate the derivative of a product and quotient of two polynomials.
Preview Activity 2.3. Let f and g be the functions defined by f (t) = 2t 2 and g(t) = t 3 +4t.
(a) Determine f ′ (t) and g ′ (t).
(b) Let p(t) = 2t 2 (t 3 + 4t) and observe that p(t) = f (t) · g(t). Rewrite the formula for
p by distributing the 2t 2 term. Then, compute p ′ (t) using the sum and constant
multiple rules.
(c) True or false: p ′ (t) = f ′ (t) · g ′ (t).
(d) Let q(t) =
t 3 + 4t
2t 2 and observe that q(t) =
g(t)
f (t)
. Rewrite the formula for q by
dividing each term in the numerator by the denominator and simplify to write q
as a sum of constant multiples of powers of t. Then, compute q ′ (t) using the sum
and constant multiple rules.
(e) True or false: q
′ (t) =
g ′ (t)
f ′ (t)
.
⊲⊳
The product rule
As parts (b) and (d) of Preview Activity 2.3 show, it is not true in general that the derivative
of a product of two functions is the product of the derivatives of those functions. Indeed, the
rule for differentiating a function of the form p(x) = f (x) · g(x) in terms of the derivatives
of f and g is more complicated than simply taking the product of the derivatives of f and
g. To see further why this is the case, as well as to begin to understand how the product
rule actually works, we consider an example involving meaningful functions.
Say that an investor is regularly purchasing stock in a particular company. Let N(t) be
a function that represents the number of shares owned on day t, where t = 0 represents
the first day on which shares were purchased. Further, let S(t) be a function that gives the
value of one share of the stock on day t; note that the units on S(t) are dollars per share.
Moreover, to compute the total value on day t of the stock held by the investor, we use the
function V (t) = N(t) · S(t). By taking the product
V (t) = N(t) shares · S(t) dollars per share,
we have the total value in dollars of the shares held. Observe that over time, both the
number of shares and the value of a given share will vary. The derivative N ′ (t) measures
the rate at which the number of shares held is changing, while S ′ (t) measures the rate at