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
![### Finding the First Partial Derivatives of a Function
Given the function:
\[ w = \frac{e^v}{u + v^8} \]
We want to find its first partial derivatives.
1. **Partial Derivative with Respect to \( u \)**:
\[ \frac{\partial w}{\partial u} = \boxed{\frac{-e^v}{(u+v^2)^2}} \]
However, this answer is marked with a red cross, indicating it might be incorrect.
2. **Partial Derivative with Respect to \( v \)**:
\[ \frac{\partial w}{\partial v} = \] (expression to be determined)
To find the first partial derivative of \( w \) with respect to \( v \), we recognize that \( w \) is a function of both \( u \) and \( v \), and apply the appropriate differentiation rules.
#### Explanation of Graphs and Diagrams:
- The image contains two boxes for the partial derivatives of \( w \) with respect to \( u \) and \( v \).
- The box for the partial derivative with respect to \( u \) is filled with the expression \(\frac{-e^v}{(u+v^2)^2}\) and marked with a red cross.
- The box for the partial derivative with respect to \( v \) is empty, indicating it is yet to be determined.
In summary, the solution is in progress, with one partial derivative given, albeit potentially incorrect, and the other pending calculation. This educational setting emphasizes accurately solving for partial derivatives step-by-step.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F97b7422a-52f7-4bc4-acdc-28b7d7bc43e4%2F2c2eb8cc-cbdc-4336-866b-88b1b62fee4c%2F0dq6n5_processed.png&w=3840&q=75)
Transcribed Image Text:### Finding the First Partial Derivatives of a Function
Given the function:
\[ w = \frac{e^v}{u + v^8} \]
We want to find its first partial derivatives.
1. **Partial Derivative with Respect to \( u \)**:
\[ \frac{\partial w}{\partial u} = \boxed{\frac{-e^v}{(u+v^2)^2}} \]
However, this answer is marked with a red cross, indicating it might be incorrect.
2. **Partial Derivative with Respect to \( v \)**:
\[ \frac{\partial w}{\partial v} = \] (expression to be determined)
To find the first partial derivative of \( w \) with respect to \( v \), we recognize that \( w \) is a function of both \( u \) and \( v \), and apply the appropriate differentiation rules.
#### Explanation of Graphs and Diagrams:
- The image contains two boxes for the partial derivatives of \( w \) with respect to \( u \) and \( v \).
- The box for the partial derivative with respect to \( u \) is filled with the expression \(\frac{-e^v}{(u+v^2)^2}\) and marked with a red cross.
- The box for the partial derivative with respect to \( v \) is empty, indicating it is yet to be determined.
In summary, the solution is in progress, with one partial derivative given, albeit potentially incorrect, and the other pending calculation. This educational setting emphasizes accurately solving for partial derivatives step-by-step.
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