Derivatives of Trigonometric
Functions
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Printed: May 1, 2014
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C HAPTER
Chapter 1. Derivatives of Trigonometric Functions
1
Derivatives of
Trigonometric Functions
Learning Objectives
A student will be able to:
• Compute the derivatives of various trigonometric functions.
If the angle h is measured in radians,
h
limh→0 sinh h = 1 and limh→0 1−cos
= 0.
h
We can use these limits to find an expression for the derivative of the six trigonometric functions sin x, cos x, tan x, sec x, csc x,
and cot x. We first consider the problem of differentiating sin x, using the definition of the derivative.
d
sin(x + h) − sin x
[sin x] = lim
h→0
dx
h
Since
sin(α + β) = sin α cos β + cos α sin β.
The derivative becomes
d
sin x cos h + cos x sin h − sin x
[sin x] = lim
h→0
dx
h cos h − 1
sin h
= lim sin x
+ cos x
h→0
h
h
1 − cos h
sin h
= −sin x · lim
+ cos x · lim
h→0
h→0
h
h
= −sin x · (0) + cos x · (1)
= cos x.
Therefore,
d
[sin x] = cos x.
dx
It will be left as an exercise to prove that
d
[cos x] = −sin x.
dx
1
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The derivatives of the remaining trigonometric functions are shown in the table below.
Derivatives of Trigonometric Functions
d
[sin x] = cos x
dx
d
[cos x] = −sin x
dx
d
[tan x] = sec2 x
dx
d
[sec x] = sec x tan x
dx
d
[csc x] = −csc x cot x
dx
d
[cot x] = −csc2 x
dx
Keep in mind that for all the derivative formulas for the trigonometric functions, the argument x is measured in
radians.
Example 1:
Show that
d
dx [tan x]
= sec2 x.
Solution:
It is possible to prove this relation by the definition of the derivative. However, we use a simpler method.
Since
tan x =
sin x
,
cos x
then
d
d sin x
[tan x] =
dx
dx cos x
Using the quotient rule,
(cos x)(cos x) − (sin x)(−sin x)
=
cos2 x
2
2
cos x + sin x
=
cos2 x
1
=
cos2 x
= sec2 x
Example 2:
Find f 0 (x) if f (x) = x2 cos x + sin x.
Solution:
Using the product rule and the formulas above, we obtain
2
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Chapter 1. Derivatives of Trigonometric Functions
f 0 (x) = x2 (−sin x) + 2x cos x + cos x
= −x2 sin x + 2x cos x + cos x.
Example 3:
Find dy/dx if y =
cos x
1−tan x
. What is the slope of the tangent line at x = π/3?
Solution:
Using the quotient rule and the formulas above, we obtain
dy (1 − tan x)(−sin x) − (cos x)(−sec2 x)
=
dx
(1 − tan x)2
−sin x + tan x sin x + cos x sec2 x
=
(1 − tan x)2
−sin x + tan x sin x + sec x
=
(1 − tan x)2
To calculate the slope of the tangent line, we simply substitute x = π/3:
dy dx =
x= π3
−sin( π3 ) + tan( π3 ) sin( π3 ) + sec( π3 )
.
(1 − tan( π3 ))2
We finally get the slope to be approximately
dy dx = 4.9.
x= π3
Example 4:
If y = sec x, find y00 (π/3).
Solution:
y0 = sec x tan x
y00 = sec x(sec2 x) + (sec x tan x) tan x
= sec3 x + sec x tan2 x.
Substituting for x = π/3,
π
π
+ sec
tan2
3
3
3
√
= (2)3 + (2)( 3)2
y00 = sec3
π
= 8 + (2)(3)
= 14.
Thus y00 (π/3) = 14.
3
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Multimedia Links
For examples of finding the derivatives of trigonometric functions (4.4), see Math Video Tutorials by James Sousa,
The Derivative of Sine and Cosine (9:21).
MEDIA
Click image to the left for more content.
Review Questions
Find the derivative y0 of the following functions:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
4
y = x sin x + 2
y = x2 cos x − x tan x − 1
y = sin2 x
x−1
y = sin
sin x+1
x+sin x
y = cos
cos
√ x−sin x
x
y = tan x + 2
y = csc x sin x + x
sec x
y = csc
x
If y = csc x, find y00 (π/6).
Use the definition of the derivative to prove that
d
dx [cos x]
= − sin x.