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Sullivan 04 apcalc4e 45342 ch02 166 233 5pp August 7, 2023 12:54

Section 2.5 • The Derivative of the Trigonometric Functions 221

EXAMPLE 4 Differentiating y = tan x

Show that the derivative of y = tan x is

d
′ 2
y = tan x = sec x
dx
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Solution
d d

sin x (cos x) − (sin x) cos x
d d sin x dx dx
′
y = tan x = =
2
dx ↑ dx cos x ↑ cos x
Identity Quotient Rule
2
2
cos x · cos x − sin x · (−sin x) cos x + sin x 1 2
= = = = sec x
2
2
2
cos x cos x cos x
R
NOW WORK Problem 15 and AP Practice Problems 3 and 7.
Table 4 lists the derivatives of the six trigonometric functions along with the domain
of each derivative.
TABLE 4
Derivative Function Domain of the Derivative Function
d
sin x = cos x (−∞, ∞)
dx
d
cos x = −sin x (−∞, ∞)
dx
d 2k + 1
2
tan x = sec x x|x 6= π, k an integer
dx 2
NOTE If the trigonometric function begins d 2
with the letter c, that is, cosine, cotangent, dx cot x = −csc x {x|x 6= kπ, k an integer}
or cosecant, then its derivative has a
minus sign. d csc x = −csc x cot x {x|x 6= kπ, k an integer}
dx

d 2k + 1
sec x = sec x tan x x|x 6= π, k an integer
dx 2
NOW WORK Problem 35.

EXAMPLE 5 Finding the Second Derivative of a Trigonometric Function
π

Find f ′′ if f (x) = sec x.
4
Solution
If f (x) = sec x, then f (x) = sec x tan x and
′
d d d
′′
f (x) = (sec x tan x) = (sec x) tan x + sec x (tan x)
dx ↑ dx dx
Use the Product Rule.
2 3 2
= sec x · sec x + (sec x tan x) tan x = sec x + sec x tan x
π 3 π π 2 π 2
√ 3 √ √ √ √
f ′′ = sec + sec tan = 2 + 2 · 1 = 2 2 + 2 = 3 2
4 4 4 4 ↑
π √ π
sec = 2; tan = 1
4 4
R
NOW WORK Problem 45 and AP Practice Problems 4, 5, and 11.
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