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Solution
a. To differentiate this function, use the chain rule.
y cos 3x
dy d1cos 3x2
dx d13x2
sin 3x 132
3 sin 3x
b. To find the derivative, use the product rule.
y x sin x
d13x2
dx
dy
dx dx
d1sin x2
sin x x
dx dx
112 sin x x cos x
sin x x cos x
(Chain rule)
(Product rule)
EXAMPLE 2
Reasoning about the derivatives of sinusoidal functions
dy
Determine for each function.
dx
a. y sin x 2
b. y sin 2 x
Solution
a. To differentiate this composite function, use the chain rule and change of
variable.
Here, the inner function is u x 2 , and the outer function is y sin u.
dy
Then,
dx dy du
(Chain rule)
du dx
1cos u212x2
2x cos x 2
b. Since y sin 2 x 1sin x2 2 , we use the chain rule with y u 2 , where
u sin x.
dy
Then,
dx dy du
(Chain rule)
du dx
12u21cos x2
2 sin x cos x
(Substitute)
(Substitute)
252
5.4 THE DERIVATIVES OF y sin x AND y cos x
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