Question

If $H(\theta)=\theta \sin \theta,$ find $H^{\prime}(\theta)$ and $H^{\pi}(\theta)$.

Name: Problem 27, Chapter 3: Calculus
Uploaded: 2021-08-24T18:32:04-08:00
Duration: 1 min 44 s
Channel: Lauren Shelton
Description: If H(θ)=θsinθ, find H^'(θ) and H^π(θ).

   If $H(\theta)=\theta \sin \theta,$ find $H^{\prime}(\theta)$ and $H^{\pi}(\theta)$.

Calculus

James Stewart 4th Edition

Chapter 3, Problem 27

Instant Answer

Solved by Expert Lauren Shelton

08/24/2021

Step 1

We need to find the first derivative of this function. We can use the product rule for differentiation which states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times Show more…

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If $H(\theta)=\theta \sin \theta,$ find $H^{\prime}(\theta)$ and $H^{\pi}(\theta)$.

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Key Concepts

Product Rule

The product rule is a fundamental differentiation technique used when differentiating a product of two functions. It states that the derivative of a product f(x)g(x) is given by f?'(x)g(x) + f(x)g?'(x). This rule is essential for finding H?(?) when H(?) is given as a product of functions.

Differentiation of Trigonometric Functions

Differentiating trigonometric functions involves knowing the standard derivatives of functions like sin?? and cos??. In this context, the derivative of sin?? is cos??, which is used in applying the product rule to differentiate H(?) = ? sin??.

Function Exponentiation

Function exponentiation involves raising the output of a function to a given power. When a function H(?) is expressed as H^(?)(?), it denotes the value of H(?) raised to the ? power, namely (? sin??)^?. This concept is separate from taking derivatives and relies on the rules of exponents applied to the function's value.

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