   Chapter 3.3, Problem 15E ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343

#### Solutions

Chapter
Section ### Single Variable Calculus: Early Tr...

8th Edition
James Stewart
ISBN: 9781305270343
Textbook Problem

# Differentiate.f(θ) = θ cos θ sin θ

To determine

To find: The differentiation of f(θ)=θcosθsinθ.

Explanation

Given:

The function is f(θ)=θcosθsinθ.

Formula used:

Product Rule:

If f1(θ) and f2(θ) are both differentiable functions, then the product rule is,

ddθ[f1(θ)f2(θ)]=f1(θ)ddθ[f2(θ)]+f2(θ)ddθ[f1(θ)] (1)

Calculation:

Apply the product rule as shown in equation (1).

Substitute θcosθ for f1(θ) and sinθ for f2(θ) in equation (1).

ddθ[θcosθsinθ]=θcosθddθ[sinθ]+sinθddθ[θcosθ]

Apply the product rule as shown in equation (1).

ddθ[θcosθsinθ]=θcosθddθ[sinθ]+sinθ[θddθ[cosθ]+cosθddθ[θ]]                        =θcosθ(cosθ)+sinθ[θ(sinθ)+cosθ(1)]                       =θcos2θθsin2θ+sin

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