2.4. DERIVATIVES OF OTHER TRIGONOMETRIC FUNCTIONS
117
a damped oscillation. Suppose that for a particular object, its displacement from
equilibrium (where the object sits at rest) is modeled by the function
s(t) =
15 sin(t)
e t
.
Assume that s is measured in inches and t in seconds. Sketch a graph of
this function for t ≥ 0 to see how it represents the situation described. Then
compute ds/dt, state the units on this function, and explain what it tells you
about the object’s motion. Finally, compute and interpret s ′ (2).
⊳
Summary
In this section, we encountered the following important ideas:
• The derivatives of the other four trigonometric functions are
d
dx
[tan(x)] = sec
2 (x),
d
dx
[cot(x)] = − csc
2 (x),
d
dx
[sec(x)] = sec(x) tan(x), and
d
dx
[csc(x)] = − csc(x) cot(x).
Each derivative exists and is defined on the same domain as the original function. For
example, both the tangent function and its derivative are defined for all real numbers x
such that x
kπ
2 , where k = ±1, ±2, . . ..
• The above four rules for the derivatives of the tangent, cotangent, secant, and cosecant
can be used along with the rules for power functions, exponential functions, and the
sine and cosine, as well as the sum, constant multiple, product, and quotient rules, to
quickly differentiate a wide range of different functions.
Exercises
1. An object moving vertically has its height at time t (measured in feet, with time in
seconds) given by the function h(t) = 3 +
2 cos(t)
1.2 t .
(a) What is the object’s instantaneous velocity when t = 2?
(b) What is the object’s acceleration at the instant t = 2?
(c) Describe in everyday language the behavior of the object at the instant t = 2.
2. Let f (x) = sin(x) cot(x).
(a) Use the product rule to find f ′ (x).
117
a damped oscillation. Suppose that for a particular object, its displacement from
equilibrium (where the object sits at rest) is modeled by the function
s(t) =
15 sin(t)
e t
.
Assume that s is measured in inches and t in seconds. Sketch a graph of
this function for t ≥ 0 to see how it represents the situation described. Then
compute ds/dt, state the units on this function, and explain what it tells you
about the object’s motion. Finally, compute and interpret s ′ (2).
⊳
Summary
In this section, we encountered the following important ideas:
• The derivatives of the other four trigonometric functions are
d
dx
[tan(x)] = sec
2 (x),
d
dx
[cot(x)] = − csc
2 (x),
d
dx
[sec(x)] = sec(x) tan(x), and
d
dx
[csc(x)] = − csc(x) cot(x).
Each derivative exists and is defined on the same domain as the original function. For
example, both the tangent function and its derivative are defined for all real numbers x
such that x
kπ
2 , where k = ±1, ±2, . . ..
• The above four rules for the derivatives of the tangent, cotangent, secant, and cosecant
can be used along with the rules for power functions, exponential functions, and the
sine and cosine, as well as the sum, constant multiple, product, and quotient rules, to
quickly differentiate a wide range of different functions.
Exercises
1. An object moving vertically has its height at time t (measured in feet, with time in
seconds) given by the function h(t) = 3 +
2 cos(t)
1.2 t .
(a) What is the object’s instantaneous velocity when t = 2?
(b) What is the object’s acceleration at the instant t = 2?
(c) Describe in everyday language the behavior of the object at the instant t = 2.
2. Let f (x) = sin(x) cot(x).
(a) Use the product rule to find f ′ (x).