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4. If y = tan-¹(cos x), then dy = dx

4. If y = tan-¹(cos x), then dy = dx

Calculus: Early Transcendentals
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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AP #4

## Calculus Problem ### Problem Statement 4. If \( y = \tan^{-1} (\cos x) \), then \(\frac{dy}{dx} =\) ### Instructions To solve this problem, differentiate the function \( y = \tan^{-1} (\cos x) \) with respect to \( x \) to find the value of \(\frac{dy}{dx}\). ### Solution Approach 1. **Differentiate using the chain rule:** - Recognize that \( y = \tan^{-1}(u) \) where \( u = \cos x \). - Use the chain rule: \(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\). 2. **Differentiate the outer function:** - \(\frac{dy}{du} = \frac{1}{1 + u^2}\). 3. **Differentiate the inner function:** - \( u = \cos x \) implies \(\frac{du}{dx} = -\sin x\). 4. **Combine the results:** - \(\frac{dy}{dx} = \frac{1}{1 + \cos^2 x} \cdot (-\sin x)\). 5. **Simplify if necessary:** - Simplify the expression to obtain the final derivative result. This detailed approach will guide students on how to methodically find the derivative using calculus rules.
Transcribed Image Text:## Calculus Problem ### Problem Statement 4. If \( y = \tan^{-1} (\cos x) \), then \(\frac{dy}{dx} =\) ### Instructions To solve this problem, differentiate the function \( y = \tan^{-1} (\cos x) \) with respect to \( x \) to find the value of \(\frac{dy}{dx}\). ### Solution Approach 1. **Differentiate using the chain rule:** - Recognize that \( y = \tan^{-1}(u) \) where \( u = \cos x \). - Use the chain rule: \(\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}\). 2. **Differentiate the outer function:** - \(\frac{dy}{du} = \frac{1}{1 + u^2}\). 3. **Differentiate the inner function:** - \( u = \cos x \) implies \(\frac{du}{dx} = -\sin x\). 4. **Combine the results:** - \(\frac{dy}{dx} = \frac{1}{1 + \cos^2 x} \cdot (-\sin x)\). 5. **Simplify if necessary:** - Simplify the expression to obtain the final derivative result. This detailed approach will guide students on how to methodically find the derivative using calculus rules.
Expert Solution
Step 1

As we know the Chain rule:

fgx'=f'gx·g'x

And

Use the common derivatives.

ddxtan-1x=1x2+1ddxcosx=-sinx

Given:

y=tan-1cosx

steps

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