Find the derivative of f (x) = ln O f'(x) © f'(x) O f'(x) O f'(x) = = = = | 8 + sin x f (x) = ln ( 22 +cot x ²/1 + tan x + +cot + 2 2x+1 2 2x+1 2 2x+1 2 2x+1 x² sin x 2x+1
Find the derivative of f (x) = ln O f'(x) © f'(x) O f'(x) O f'(x) = = = = | 8 + sin x f (x) = ln ( 22 +cot x ²/1 + tan x + +cot + 2 2x+1 2 2x+1 2 2x+1 2 2x+1 x² sin x 2x+1
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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![**Problem Statement:**
Find the derivative of \( f(x) = \ln \left( \frac{x^2 \sin x}{2x+1} \right) \).
**Options:**
1. \( f'(x) = \frac{2}{x} + \sin x - \frac{2}{2x+1} \)
2. \( f'(x) = \frac{2}{x} + \cot x - \frac{2}{2x+1} \) (Correct answer)
3. \( f'(x) = \frac{2}{x} + \tan x + \frac{2}{2x+1} \)
4. \( f'(x) = \frac{1}{x} + \cot x + \frac{2}{2x+1} \)
**Explanation for the Correct Option:**
The derivative of a logarithmic function of the form \( \ln\left(\frac{u}{v}\right) \) can be found using the formula:
\[
f'(x) = \frac{u'}{u} - \frac{v'}{v}
\]
where \( u = x^2 \sin x \) and \( v = 2x + 1 \).
- The derivative \( u' = \frac{d}{dx}(x^2 \sin x) \) can be solved using product rule.
- The derivative \( v' = \frac{d}{dx}(2x+1) \).
Substituting these into the formula will provide the correct derivative as shown in option 2.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7353de8a-3834-465a-af3b-558129de7e04%2F8c3d3547-b121-47af-acca-398e2a28b790%2Fqaixbqj_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the derivative of \( f(x) = \ln \left( \frac{x^2 \sin x}{2x+1} \right) \).
**Options:**
1. \( f'(x) = \frac{2}{x} + \sin x - \frac{2}{2x+1} \)
2. \( f'(x) = \frac{2}{x} + \cot x - \frac{2}{2x+1} \) (Correct answer)
3. \( f'(x) = \frac{2}{x} + \tan x + \frac{2}{2x+1} \)
4. \( f'(x) = \frac{1}{x} + \cot x + \frac{2}{2x+1} \)
**Explanation for the Correct Option:**
The derivative of a logarithmic function of the form \( \ln\left(\frac{u}{v}\right) \) can be found using the formula:
\[
f'(x) = \frac{u'}{u} - \frac{v'}{v}
\]
where \( u = x^2 \sin x \) and \( v = 2x + 1 \).
- The derivative \( u' = \frac{d}{dx}(x^2 \sin x) \) can be solved using product rule.
- The derivative \( v' = \frac{d}{dx}(2x+1) \).
Substituting these into the formula will provide the correct derivative as shown in option 2.
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