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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Please explain clearly, no cursive writing.
![### Problem Statement
Compute the partial derivatives of the function \(9e^{8x} \sin(7y)\) with respect to \(x\) and \(y\).
#### Given Formula:
\[
\frac{\partial}{\partial x} 9e^{8x} \sin(7y) = \underline{\quad\quad\quad}
\]
\[
\frac{\partial}{\partial y} 9e^{8x} \sin(7y) = \underline{\quad\quad\quad}
\]
### Explanation
1. **Partial Derivative with respect to \(x\):**
- Treat \(y\) as a constant.
- Differentiate \(9e^{8x}\) using the chain rule.
- Result: \(72e^{8x} \sin(7y)\)
2. **Partial Derivative with respect to \(y\):**
- Treat \(x\) as a constant.
- Differentiate \(\sin(7y)\) using the chain rule, yielding \(7\cos(7y)\).
- Result: \(63e^{8x} \cos(7y)\)
### Answers
\[
\frac{\partial}{\partial x} 9e^{8x} \sin(7y) = 72e^{8x} \sin(7y)
\]
\[
\frac{\partial}{\partial y} 9e^{8x} \sin(7y) = 63e^{8x} \cos(7y)
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcb9da028-c1e8-4fb4-9df2-7f4287e8030e%2F4d786306-c77e-45dd-b93c-89d407e4e2a0%2Fmssilp6_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Compute the partial derivatives of the function \(9e^{8x} \sin(7y)\) with respect to \(x\) and \(y\).
#### Given Formula:
\[
\frac{\partial}{\partial x} 9e^{8x} \sin(7y) = \underline{\quad\quad\quad}
\]
\[
\frac{\partial}{\partial y} 9e^{8x} \sin(7y) = \underline{\quad\quad\quad}
\]
### Explanation
1. **Partial Derivative with respect to \(x\):**
- Treat \(y\) as a constant.
- Differentiate \(9e^{8x}\) using the chain rule.
- Result: \(72e^{8x} \sin(7y)\)
2. **Partial Derivative with respect to \(y\):**
- Treat \(x\) as a constant.
- Differentiate \(\sin(7y)\) using the chain rule, yielding \(7\cos(7y)\).
- Result: \(63e^{8x} \cos(7y)\)
### Answers
\[
\frac{\partial}{\partial x} 9e^{8x} \sin(7y) = 72e^{8x} \sin(7y)
\]
\[
\frac{\partial}{\partial y} 9e^{8x} \sin(7y) = 63e^{8x} \cos(7y)
\]
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