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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![The task is to find the derivative of the function \( y = \cot(4 \cdot x) \).
### Problem Statement:
Given:
\[ y = \cot(4 \cdot x) \]
Find:
\[ \frac{dy}{dx} \]
### Solution Approach:
To determine the derivative, we use the chain rule. The derivative of \( \cot(u) \) with respect to \( u \) is \(-\csc^2(u)\). For this problem, \( u = 4x \).
1. Differentiate \( y = \cot(4x) \).
\[
\frac{dy}{dx} = \frac{d}{dx} [\cot(4x)] = -\csc^2(4x) \cdot \frac{d}{dx} [4x]
\]
2. Compute the derivative of \( 4x \).
\[
\frac{d}{dx} [4x] = 4
\]
3. Substitute back into the equation:
\[
\frac{dy}{dx} = -4\csc^2(4x)
\]
### Final Result:
\[ \frac{dy}{dx} = -4\csc^2(4x) \]
This concludes the differentiation of the given function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3aff424b-0e96-4e90-b1d6-dc40303839e0%2F6add9670-2539-47fa-bac2-259fc6536207%2F1bov8x8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The task is to find the derivative of the function \( y = \cot(4 \cdot x) \).
### Problem Statement:
Given:
\[ y = \cot(4 \cdot x) \]
Find:
\[ \frac{dy}{dx} \]
### Solution Approach:
To determine the derivative, we use the chain rule. The derivative of \( \cot(u) \) with respect to \( u \) is \(-\csc^2(u)\). For this problem, \( u = 4x \).
1. Differentiate \( y = \cot(4x) \).
\[
\frac{dy}{dx} = \frac{d}{dx} [\cot(4x)] = -\csc^2(4x) \cdot \frac{d}{dx} [4x]
\]
2. Compute the derivative of \( 4x \).
\[
\frac{d}{dx} [4x] = 4
\]
3. Substitute back into the equation:
\[
\frac{dy}{dx} = -4\csc^2(4x)
\]
### Final Result:
\[ \frac{dy}{dx} = -4\csc^2(4x) \]
This concludes the differentiation of the given function.
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