   Chapter 8.4, Problem 37E ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

#### Solutions

Chapter
Section ### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Differentiating Trigonometric Functions In Exercises 29-40, find the derivative of the function and simplify your answer by using the trigonometric identities listed in Section 8.2. y = sin 3 x 3 − sin 5 x 5

To determine

To calculate: The simplified derivative of the trigonometric function y=sin3x3sin5x5 using trigonometric identities.

Explanation

Given Information:

The trigonometric function:

y=sin3x3sin5x5

Formula used:

Sine differentiation rule:

ddx[sinu]=cosududx

General power rule of differentiation:

ddx[xn]=nxn1

The sum and difference formula of differentiation:

ddx[f(x)+g(x)]=f'(x)+g'(x)

Pythagorean identity:

cos2θ=1sin2θ

Calculation:

Consider the trigonometric function:

y=sin3x3sin5x5

Apply the sum and difference formula of differentiation on the above function.

dydx=ddx[sin3x3]ddx[sin5x5]

Factor out the real numbers.

ddx[sin3x3]ddx[sin5x5]=13ddx[sin3x]15ddx[sin5x]

Now, apply the general power rule and sine rule of differentiation

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