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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 219

y
3π π 5π
f ( ) ! 0 f ( ) ! 0 f ( ) ! 0
2 2 2
1
3π 5π 3π π π (0, 0) π π 3π 2π 5π 3π x
  
2 2 2 2 2 2
1
5π
π
3π
f ( ) ! 0 f ( ) ! 0 f ( ) ! 0
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2 2 2
f(x) ! sin x
y
(2π, 1) (0, 1) (2π, 1)
1
x
3π 5π 2π 3π π π π π 3π 2π 5π 3π
  
2 2 2 2 2 2
1
(3π, 1) (π, 1) (π, 1) (3π, 1)
f (x) ! cos x
Figure 30

EXAMPLE 1 Differentiating the Sine Function

Find y if:
′
sin x
2
x
(a) y = x + 4 sin x (b) y = x sin x (c) y = (d) y = e sin x
x
Solution
(a) Use the Sum Rule and the Constant Multiple Rule.
d d d d
′
y = (x + 4 sin x) = x + (4 sin x) = 1 + 4 sin x = 1 + 4 cos x
dx dx dx dx
(b) Use the Product Rule.
d d d
′ 2 2 2 2
y = (x sin x) = x sin x + x sin x = x cos x + 2x sin x
dx dx dx
(c) Use the Quotient Rule.
d d

sin x · x − sin x · x

d sin x dx dx x cos x − sin x
′
y = = =
dx x x 2 x 2
(d) Use the Product Rule.
d d d
′ x x x
y = (e sin x) = e sin x + e sin x
dx dx dx
x
x
x
= e cos x + e sin x = e (cos x + sin x)
R
NOW WORK Problems 5, 29, and AP Practice Problems 1 and 10.
THEOREM Derivative of y = cos x
The derivative of y = cos x is

d
′
y = cos x = −sin x
dx

You are asked to prove this in Problem 75.

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