2.3. THE PRODUCT AND QUOTIENT RULES
109
sum rule. Applying these rules, we now have
s
′ (y) =
(y 2 + 1)[y · 7 y ln(7) + 7 y · 1] − y · 7 y [2y]
(y 2 + 1) 2
.
While some minor simplification is possible, we are content to leave s ′ (y) in its current
form, having found the desired derivative of s. In summary, to compute the derivative of s,
we applied the quotient rule. In so doing, when it was time to compute the derivative of
the top function, we used the product rule; at the point where we found the derivative of
the bottom function, we used the sum rule.
In general, one of the main keys to success in applying derivative rules is to recognize
the structure of the function, followed by the careful and diligent application of relevant
derivative rules. The best way to get good at this process is by doing a large number of
exercises, and the next activity provides some practice and exploration to that end.
Activity 2.9.
Use relevant derivative rules to answer each of the questions below. Throughout, be
sure to use proper notation and carefully label any derivative you find by name.
(a) Let f (r) = (5r
3 + sin(r))(4
r − 2 cos(r)). Find f ′ (r).
(b) Let p(t) =
cos(t)
t 6 · 6 t . Find p ′ (t).
(c) Let g(z) = 3z
7 e
z − 2z
2 sin(z) +
z
z 2 + 1
. Find g ′ (z).
(d) A moving particle has its position in feet at time t in seconds given by the
function s(t) =
3 cos(t) − sin(t)
e t
. Find the particle’s instantaneous velocity at
the moment t = 1.
(e) Suppose that f (x) and g(x) are differentiable functions and it is known that
f (3) = −2, f ′ (3) = 7, g(3) = 4, and g ′ (3) = −1. If p(x) = f (x) · g(x) and
q(x) =
f (x)
g(x)
, calculate p ′ (3) and q ′ (3).
⊳
As the algebraic complexity of the functions we are able to differentiate continues to
increase, it is important to remember that all of the derivative’s meaning continues to hold.
Regardless of the structure of the function f , the value of f ′ (a) tells us the instantaneous
rate of change of f with respect to x at the moment x = a, as well as the slope of the
tangent line to y = f (x) at the point (a, f (a)).
109
sum rule. Applying these rules, we now have
s
′ (y) =
(y 2 + 1)[y · 7 y ln(7) + 7 y · 1] − y · 7 y [2y]
(y 2 + 1) 2
.
While some minor simplification is possible, we are content to leave s ′ (y) in its current
form, having found the desired derivative of s. In summary, to compute the derivative of s,
we applied the quotient rule. In so doing, when it was time to compute the derivative of
the top function, we used the product rule; at the point where we found the derivative of
the bottom function, we used the sum rule.
In general, one of the main keys to success in applying derivative rules is to recognize
the structure of the function, followed by the careful and diligent application of relevant
derivative rules. The best way to get good at this process is by doing a large number of
exercises, and the next activity provides some practice and exploration to that end.
Activity 2.9.
Use relevant derivative rules to answer each of the questions below. Throughout, be
sure to use proper notation and carefully label any derivative you find by name.
(a) Let f (r) = (5r
3 + sin(r))(4
r − 2 cos(r)). Find f ′ (r).
(b) Let p(t) =
cos(t)
t 6 · 6 t . Find p ′ (t).
(c) Let g(z) = 3z
7 e
z − 2z
2 sin(z) +
z
z 2 + 1
. Find g ′ (z).
(d) A moving particle has its position in feet at time t in seconds given by the
function s(t) =
3 cos(t) − sin(t)
e t
. Find the particle’s instantaneous velocity at
the moment t = 1.
(e) Suppose that f (x) and g(x) are differentiable functions and it is known that
f (3) = −2, f ′ (3) = 7, g(3) = 4, and g ′ (3) = −1. If p(x) = f (x) · g(x) and
q(x) =
f (x)
g(x)
, calculate p ′ (3) and q ′ (3).
⊳
As the algebraic complexity of the functions we are able to differentiate continues to
increase, it is important to remember that all of the derivative’s meaning continues to hold.
Regardless of the structure of the function f , the value of f ′ (a) tells us the instantaneous
rate of change of f with respect to x at the moment x = a, as well as the slope of the
tangent line to y = f (x) at the point (a, f (a)).