2.4. DERIVATIVES OF OTHER TRIGONOMETRIC FUNCTIONS
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rule. Hence
g
′ (x) =
sin(x)(− sin(x)) − cos(x) cos(x)
sin
2 (x)
= −
sin
2 (x) + cos 2 (x)
sin
2 (x)
By the Fundamental Trigonometric Identity, we see that g ′ (x) = −
1
sin
2 (x)
; recalling that
csc(x) =
1
sin(x) , it follows that we can most simply express g ′ by the rule
g
′ (x) = − csc
2 (x).
Note that neither g nor g ′ is defined when sin(x) = 0, which occurs at every integer
multiple of π. Hence we have the following rule.
Cotangent Function: For all real numbers x such that x kπ, where k =
0, ±1, ±2, . . .,
d
dx
[cot(x)] = − csc
2 (x).
Observe that the shortcut rule for the cotangent function is very similar to the rule we
discovered in Preview Activity 2.4 for the tangent function.
Tangent Function: For all real numbers x such that x
(2k+1)π
2
, where k =
±1, ±2, . . .,
d
dx
[tan(x)] = sec
2 (x).
In the next two activities, we develop the rules for differentiating the secant and
cosecant functions.
Activity 2.10.
Let h(x) = sec(x) and recall that sec(x) =
1
cos(x) .
(a) What is the domain of h?
(b) Use the quotient rule to develop a formula for h ′ (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
h ′ (x) only in terms of tan(x) and sec(x)?
(d) What is the domain of h ′ ? How does this compare to the domain of h?
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