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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Question
![**Problem Statement:**
Find the derivative \( y = e^{\cot x} \).
**Answer Choices:**
1. \(\frac{dy}{dx} = -e^{\cot x} \csc^2 x\)
2. \(\frac{dy}{dx} = \cot x \csc^2 x\)
3. \(\frac{dy}{dx} = e^{\cot x} \csc^2 x\)
4. \(\frac{dy}{dx} = -e^{\cot x}\)
**Explanation:**
To find the correct derivative, apply the chain rule. The derivative of the \(e^{u}\) function is \(e^{u} \cdot \frac{du}{dx}\), where \(u = \cot x\).
The derivative of \(\cot x\) is \(-\csc^2 x\), so:
\[
\frac{dy}{dx} = e^{\cot x} \cdot (-\csc^2 x) = -e^{\cot x} \csc^2 x
\]
Therefore, the correct answer is:
1. \(\frac{dy}{dx} = -e^{\cot x} \csc^2 x\)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8212daa0-4bcd-44c2-b8bb-69661bb0db17%2F2c84989d-8be1-46e2-85d0-c86b40a06861%2Fvzlilyb_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the derivative \( y = e^{\cot x} \).
**Answer Choices:**
1. \(\frac{dy}{dx} = -e^{\cot x} \csc^2 x\)
2. \(\frac{dy}{dx} = \cot x \csc^2 x\)
3. \(\frac{dy}{dx} = e^{\cot x} \csc^2 x\)
4. \(\frac{dy}{dx} = -e^{\cot x}\)
**Explanation:**
To find the correct derivative, apply the chain rule. The derivative of the \(e^{u}\) function is \(e^{u} \cdot \frac{du}{dx}\), where \(u = \cot x\).
The derivative of \(\cot x\) is \(-\csc^2 x\), so:
\[
\frac{dy}{dx} = e^{\cot x} \cdot (-\csc^2 x) = -e^{\cot x} \csc^2 x
\]
Therefore, the correct answer is:
1. \(\frac{dy}{dx} = -e^{\cot x} \csc^2 x\)
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