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2.4. DERIVATIVES OF OTHER TRIGONOMETRIC FUNCTIONS
Activity 2.11.
Let p(x) = csc(x) and recall that csc(x) =
1
sin(x) .
(a) What is the domain of p?
(b) Use the quotient rule to develop a formula for p ′ (x) that is expressed completely
in terms of sin(x) and cos(x).
(c) How can you use other relationships among trigonometric functions to write
p ′ (x) only in terms of cot(x) and csc(x)?
(d) What is the domain of p ′ ? How does this compare to the domain of p?
⊳
The quotient rule has thus enabled us to determine the derivatives of the tangent,
cotangent, secant, and cosecant functions, expanding our overall library of basic functions
we can differentiate. Moreover, we observe that just as the derivative of any polynomial
function is a polynomial, and the derivative of any exponential function is another
exponential function, so it is that the derivative of any basic trigonometric function is
another function that consists of basic trigonometric functions. This makes sense because
all trigonometric functions are periodic, and hence their derivatives will be periodic, too.
As has been and will continue to be the case throughout our work in Chapter 2, the
derivative retains all of its fundamental meaning as an instantaneous rate of change and
as the slope of the tangent line to the function under consideration. Our present work
primarily expands the list of functions for which we can quickly determine a formula for
the derivative. Moreover, with the addition of tan(x), cot(x), sec(x), and csc(x) to our
library of basic functions, there are many more functions we can differentiate through the
sum, constant multiple, product, and quotient rules.
Activity 2.12.
Answer each of the following questions. Where a derivative is requested, be sure to
label the derivative function with its name using proper notation.
(a) Let f (x) = 5 sec(x) − 2 csc(x). Find the slope of the tangent line to f at the
point where x =
π
3 .
(b) Let p(z) = z 2 sec(z) − z cot(z). Find the instantaneous rate of change of p at
the point where z =
π
4 .
(c) Let h(t) =
tan(t)
t 2 + 1
− 2e
t cos(t). Find h ′ (t).
(d) Let g(r) =
r sec(r)
5 r . Find g ′ (r).
(e) When a mass hangs from a spring and is set in motion, the object’s position
oscillates in a way that the size of the oscillations decrease. This is usually called
2.4. DERIVATIVES OF OTHER TRIGONOMETRIC FUNCTIONS
Activity 2.11.
Let p(x) = csc(x) and recall that csc(x) =
1
sin(x) .
(a) What is the domain of p?
(b) Use the quotient rule to develop a formula for p ′ (x) that is expressed completely
in terms of sin(x) and cos(x).
(c) How can you use other relationships among trigonometric functions to write
p ′ (x) only in terms of cot(x) and csc(x)?
(d) What is the domain of p ′ ? How does this compare to the domain of p?
⊳
The quotient rule has thus enabled us to determine the derivatives of the tangent,
cotangent, secant, and cosecant functions, expanding our overall library of basic functions
we can differentiate. Moreover, we observe that just as the derivative of any polynomial
function is a polynomial, and the derivative of any exponential function is another
exponential function, so it is that the derivative of any basic trigonometric function is
another function that consists of basic trigonometric functions. This makes sense because
all trigonometric functions are periodic, and hence their derivatives will be periodic, too.
As has been and will continue to be the case throughout our work in Chapter 2, the
derivative retains all of its fundamental meaning as an instantaneous rate of change and
as the slope of the tangent line to the function under consideration. Our present work
primarily expands the list of functions for which we can quickly determine a formula for
the derivative. Moreover, with the addition of tan(x), cot(x), sec(x), and csc(x) to our
library of basic functions, there are many more functions we can differentiate through the
sum, constant multiple, product, and quotient rules.
Activity 2.12.
Answer each of the following questions. Where a derivative is requested, be sure to
label the derivative function with its name using proper notation.
(a) Let f (x) = 5 sec(x) − 2 csc(x). Find the slope of the tangent line to f at the
point where x =
π
3 .
(b) Let p(z) = z 2 sec(z) − z cot(z). Find the instantaneous rate of change of p at
the point where z =
π
4 .
(c) Let h(t) =
tan(t)
t 2 + 1
− 2e
t cos(t). Find h ′ (t).
(d) Let g(r) =
r sec(r)
5 r . Find g ′ (r).
(e) When a mass hangs from a spring and is set in motion, the object’s position
oscillates in a way that the size of the oscillations decrease. This is usually called