2.3. THE PRODUCT AND QUOTIENT RULES
107
the bottom, all over the bottom squared.” For example, if Q(t) = sin(t)/2 t , then we can
identify the top function as sin(t) and the bottom function as 2 t . By the quotient rule, we
then have that Q ′ will be given by the bottom, 2 t , times the derivative of the top, cos(t),
minus the top, sin(t), times the derivative of the bottom, 2 t ln(2), all over the bottom
squared, (2 t ) 2 . That is,
Q
′ (t) =
2 t cos(t) − sin(t)2 t ln(2)
(2 t ) 2
.
In this particular example, it is possible to simplify Q ′ (t) by removing a factor of 2 t from
both the numerator and denominator, hence finding that
Q
′ (t) =
cos(t) − sin(t) ln(2)
2 t
.
In general, we must be careful in doing any such simplification, as we don’t want to
correctly execute the quotient rule but then find an incorrect overall derivative due to an
algebra error. As such, we will often place more emphasis on correctly using derivative
rules than we will on simplifying the result that follows. The next activity further explores
the use of the quotient rule.
Activity 2.8.
Use the quotient rule to answer each of the questions below. Throughout, be sure to
carefully label any derivative you find by name. That is, if you’re given a formula for
f (x), clearly label the formula you find for f ′ (x). It is not necessary to algebraically
simplify any of the derivatives you compute.
(a) Let r(z) =
3 z
z 4 + 1
. Find r ′ (z).
(b) Let v(t) =
sin(t)
cos(t) + t 2 . Find v ′ (t).
(c) Determine the slope of the tangent line to the curve R(x) =
x 2 − 2x − 8
x 2 − 9
at the
point where x = 0.
(d) When a camera flashes, the intensity I of light seen by the eye is given by the
function
I(t) =
100t
e t ,
where I is measured in candles and t is measured in milliseconds. Compute
I ′ (0.5), I ′ (2), and I ′ (5); include appropriate units on each value; and discuss
the meaning of each.
⊳
107
the bottom, all over the bottom squared.” For example, if Q(t) = sin(t)/2 t , then we can
identify the top function as sin(t) and the bottom function as 2 t . By the quotient rule, we
then have that Q ′ will be given by the bottom, 2 t , times the derivative of the top, cos(t),
minus the top, sin(t), times the derivative of the bottom, 2 t ln(2), all over the bottom
squared, (2 t ) 2 . That is,
Q
′ (t) =
2 t cos(t) − sin(t)2 t ln(2)
(2 t ) 2
.
In this particular example, it is possible to simplify Q ′ (t) by removing a factor of 2 t from
both the numerator and denominator, hence finding that
Q
′ (t) =
cos(t) − sin(t) ln(2)
2 t
.
In general, we must be careful in doing any such simplification, as we don’t want to
correctly execute the quotient rule but then find an incorrect overall derivative due to an
algebra error. As such, we will often place more emphasis on correctly using derivative
rules than we will on simplifying the result that follows. The next activity further explores
the use of the quotient rule.
Activity 2.8.
Use the quotient rule to answer each of the questions below. Throughout, be sure to
carefully label any derivative you find by name. That is, if you’re given a formula for
f (x), clearly label the formula you find for f ′ (x). It is not necessary to algebraically
simplify any of the derivatives you compute.
(a) Let r(z) =
3 z
z 4 + 1
. Find r ′ (z).
(b) Let v(t) =
sin(t)
cos(t) + t 2 . Find v ′ (t).
(c) Determine the slope of the tangent line to the curve R(x) =
x 2 − 2x − 8
x 2 − 9
at the
point where x = 0.
(d) When a camera flashes, the intensity I of light seen by the eye is given by the
function
I(t) =
100t
e t ,
where I is measured in candles and t is measured in milliseconds. Compute
I ′ (0.5), I ′ (2), and I ′ (5); include appropriate units on each value; and discuss
the meaning of each.
⊳