3.44 help pls **Problem: Find the derivative of the function.**Given the function:\[ f(\theta) = \cos(\theta^2) \]Find $ f'(\theta) = $ **Solution Approach:**To find the derivative of the given function $ f(\theta) = \cos(\theta^2) $, we will use the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.Steps:1. Identify the outer function and the inner function. - Outer function: $ \cos(u) $ - Inner function: $ u = \theta^2 $2. Differentiate the outer function with respect to $ u $: - $ \frac{d}{du}[\cos(u)] = -\sin(u) $3. Differentiate the inner function with respect to $ \theta $: - $ \frac{d}{d\theta}[\theta^2] = 2\theta $4. Apply the chain rule: - $ f'(\theta) = -\sin(\theta^2) \cdot 2\theta $Therefore, the derivative is:\[ f'(\theta) = -2\theta \sin(\theta^2) \]

Answered: Find the derivative of the function.…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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3.44 help pls

**Problem: Find the derivative of the function.**

Given the function:

\[ f(\theta) = \cos(\theta^2) \]

Find $ f'(\theta) = $

**Solution Approach:**

To find the derivative of the given function $ f(\theta) = \cos(\theta^2) $, we will use the chain rule. The chain rule states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.

Steps:
1. Identify the outer function and the inner function.
- Outer function: $ \cos(u) $
- Inner function: $ u = \theta^2 $

2. Differentiate the outer function with respect to $ u $:
- $ \frac{d}{du}[\cos(u)] = -\sin(u) $

3. Differentiate the inner function with respect to $ \theta $:
- $ \frac{d}{d\theta}[\theta^2] = 2\theta $

4. Apply the chain rule:
- $ f'(\theta) = -\sin(\theta^2) \cdot 2\theta $

Therefore, the derivative is:

\[ f'(\theta) = -2\theta \sin(\theta^2) \]

Expert Solution

Step 1: Define problem.

We have given $f open parentheses theta close parentheses equals cos open parentheses theta squared close parentheses$

We have to find the derivative of the given function.

As we know from chain rule that for a differentiable function f(g(x))-

$fraction numerator d open parentheses f open parentheses g open parentheses x close parentheses close parentheses close parentheses over denominator d x end fraction equals f apostrophe open parentheses g open parentheses x close parentheses close parentheses g apostrophe open parentheses x close parentheses$