7001_AWLThomas_ch03p122-221.qxd 10/12/09 2:22 PM Page 155
Derivatives of Trigonometric Functions
The derivative of the sine function is the cosine function:
d
ssin xd = cos x .
dx
The derivative of the cosine function is the negative of the sine function:
d
scos xd = - sin x.
dx
The derivatives of the other trigonometric functions:
d
stan xd = sec2 x
dx
d
scot xd = - csc2 x
dx
d
ssec xd = sec x tan x
dx
d
scsc xd = - csc x cot x
dx
Many phenomena of nature are approximately periodic (electromagnetic fields, heart rhythms,
tides, weather). The derivatives of sines and cosines play a key role in describing periodic
changes. This section shows how to differentiate the six basic trigonometric functions.
EXAMPLE 1
Derivative of the Sine Function
To calculate the derivative of ƒsxd= sin x , for x measured in radians, we combine the
limits with the angle sum identity for the sine function:
If ƒsxd = sin x , then sin sx + hd = sin x cos h + cos x sin h .
ƒ¿sxd = lim
h:0
= lim
h:0
ƒsx + hd - ƒsxd
sin sx + hd - sin x
= lim
h
h
h:0
ssin x cos h + cos x sin hd - sin x
sin x scos h - 1d + cos x sin h
= lim
h
h
h:0
= lim asin x
h:0
= sin x
Derivative definition
#
lim
h:0
# cos h
- 1
h
b + lim acos x
h:0
cos h - 1
+ cos x
h
(+++)+++*
limit 0
#
lim
h:0
# sin h b
h
sin h
= sin x # 0 + cos x # 1 = cos x .
h
(+)+*
limit 1
7001_AWLThomas_ch03p122-221.qxd 10/12/09 2:22 PM Page 156
EXAMPLE 2 Derivative of the Cosine Function
With the help of the angle sum formula for the cosine function,
cos sx + hd = cos x cos h - sin x sin h ,
we can compute the limit of the difference quotient:
cos sx + hd - cos x
d
scos xd = lim
dx
h
h:0
Derivative definition
= lim
scos x cos h - sin x sin hd - cos x
h
= lim
cos x scos h - 1d - sin x sin h
h
h:0
h:0
= lim cos x
# cos h
= cos x
#
- 1
h
h:0
lim
h:0
- lim sin x
# sin h
h
h:0
cos h - 1
- sin x
h
#
Cosine angle sum
identity
lim
h:0
sin h
h
= cos x # 0 - sin x # 1
= - sin x .
EXAMPLE 3 We find derivatives of the function involving differences, products, and
quotients.
(a)
y = x 2 - sin x :
dy
d
= 2x (sin x)
dx
dx
Difference Rule
= 2x - cos x
(b)
x
dy
=
dx
sin x
y =
x :
=
(c)
#
d
(sin x) - sin x # 1
dx
x2
Quotient Rule
x cos x - sin x
x2
y = sin x cos x :
dy
d
d
= sin x
(cos x) + cos x
(sin x)
dx
dx
dx
= sin x s - sin xd + cos x scos xd
= cos2 x - sin2 x
Product Rule
7001_AWLThomas_ch03p122-221.qxd 10/12/09 2:22 PM Page 157
(e)
y =
cos x
:
1 - sin x
d
d
(1 - sin x)
(cos x) - cos x
(1 - sin x)
dy
dx
dx
=
2
dx
s1 - sin xd
EXAMPLE 4
Solution
=
s1 - sin xds - sin xd - cos x s0 - cos xd
s1 - sin xd2
=
1 - sin x
s1 - sin xd2
=
1
1 - sin x
Quotient Rule
sin2 x + cos2 x = 1
Find d(tan x)> dx.
We use the Derivative Quotient Rule to calculate the derivative:
d
d
ssin xd - sin x
scos xd
dx
dx
cos2 x
cos x cos x - sin x s - sin xd
=
cos2 x
sin x
d
d
b =
stan xd =
a
dx
dx cos x
cos x
=
cos2 x + sin2 x
cos2 x
=
1
= sec2 x.
cos2 x
Quotient Rule
EXAMPLE 5
Solution
Find y– if y = sec x .
Finding the second derivative involves a combination of trigonometric
derivatives.
y = sec x
y¿ = sec x tan x
y– =
Derivative rule for secant function
d
ssec x tan xd
dx
= sec x
d
d
(tan x) + tan x
(sec x)
dx
dx
= sec x ssec2 xd + tan x ssec x tan xd
= sec3 x + sec x tan2 x
Derivative Product Rule
Derivative rules
7001_AWLThomas_ch03p122-221.qxd 10/12/09 2:22 PM Page 159
Exercises 3.5
Derivatives
In Exercises 1–18, find dy>dx.
1. y = - 10x + 3 cos x
3
2. y = x + 5 sin x
3. y = x 2 cos x
4. y = 2x sec x + 3
11. y =
1
x2
8. gsxd = csc x cot x
5. y = csc x - 4 1x + 7
cot x
1 + cot x
4
1
13. y = cos x +
tan x
6. y = x 2 cot x -
7. ƒsxd = sin x tan x
9. y = ssec x + tan xdssec x - tan xd
10. y = ssin x + cos xd sec x
12. y =
cos x
1 + sin x
In Exercises 19–22, find ds>dt.
19. s = tan t
14. y =
cos x
x
x + cos x
21. s =
1 + csc t
1 - csc t
20. s = t 2 - sec t
22. s =
sin t
1 - cos t
15. y = x 2 sin x + 2x cos x - 2 sin x
16. y = x 2 cos x - 2x sin x - 2 cos x
17. ƒsxd = x 3 sin x cos x
18. gsxd = s2 - xd tan2 x
In Exercises 23–26, find dr>du .
23. r = 4 -
u 2 sin u
24. r = u sin u + cos u
26. r = s1 + sec ud sin u
25. r = sec u csc u
35. y = sin x,
In Exercises 27–32, find dp>dq.
1
27. p = 5 + cot q
28. p = s1 + csc qd cos q
sin q + cos q
cos q
29. p =
q sin q
31. p =
q2
- 1
33. Find y– if
34. Find
=
d4
32. p =
3q + tan q
q sec q
38. x = -p>3, p>4
y = 1 + cos x, -3p>2 … x … 2p
49.
lim
u : p>6
y = sec x,
-p>2 6 x 6 p>2
x = -p>3, 3p>2
b. y = sec x .
y>dx 4
if
b. y = 9 cos x .
51. lim sec ce x + p tan a
x:0
1
1
47. lim sin a x - b
2
x:2
lim
x = -p>3, 0, p>3
37.
Trigonometric Limits
Find the limits in Exercises 47–54.
x : - p>6
x = -p, 0, 3p>2
36. y = tan x, -p>2 6 x 6 p>2
tan q
1 + tan q
a. y = -2 sin x .
48.
-3p>2 … x … 2p
30. p =
a. y = csc x .
y s4d
Tangent Lines
In Exercises 35–38, graph the curves over the given intervals, together
with their tangents at the given values of x. Label each curve and
tangent with its equation.
52. lim sin a
21 + cos sp csc xd
sin u u -
p
6
1
2
x:0
53.
50.
lim
u : p>4
tan u - 1
u - p
4
p + tan x
b
tan x - 2 sec x
lim tan a1 -
t:0
p
b - 1d
4 sec x
sin t
t b
54. lim cos a
u:0
pu
b
sin u