Use the Chain Rule to evaluate the partial derivative at the point (r, 0) = (2V2, ), where g(x, y) : -, x = 8r cos(0), x+y y = 9r sin(0). (Use symbolic notation and fractions where needed.) dg (r.0)
Use the Chain Rule to evaluate the partial derivative at the point (r, 0) = (2V2, ), where g(x, y) : -, x = 8r cos(0), x+y y = 9r sin(0). (Use symbolic notation and fractions where needed.) dg (r.0)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![**Title: Evaluating Partial Derivatives Using the Chain Rule**
**Objective:**
Learn how to evaluate the partial derivative of a function using the chain rule in the context of polar coordinates.
**Problem Statement:**
Use the Chain Rule to evaluate the partial derivative \(\frac{\partial g}{\partial \theta}\) at the point \((r, \theta) = \left(2\sqrt{2}, \frac{\pi}{4}\right)\), where
- \(g(x, y) = \frac{1}{x+y^2}\),
- \(x = 8r \cos(\theta)\),
- \(y = 9r \sin(\theta)\).
**Instructions:**
Use symbolic notation and fractions where needed to compute the solution.
**Solution:**
Find \(\frac{\partial g}{\partial \theta}\) using the chain rule and express your answer in the box provided:
\[
\frac{\partial g}{\partial \theta} \bigg|_{(r, \theta)} = \boxed{}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F51ff2e68-0bd1-44c4-8696-122fa89f1551%2Ffd145c92-e529-4cab-9bef-039a75335c95%2Fpdcdzrga_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Evaluating Partial Derivatives Using the Chain Rule**
**Objective:**
Learn how to evaluate the partial derivative of a function using the chain rule in the context of polar coordinates.
**Problem Statement:**
Use the Chain Rule to evaluate the partial derivative \(\frac{\partial g}{\partial \theta}\) at the point \((r, \theta) = \left(2\sqrt{2}, \frac{\pi}{4}\right)\), where
- \(g(x, y) = \frac{1}{x+y^2}\),
- \(x = 8r \cos(\theta)\),
- \(y = 9r \sin(\theta)\).
**Instructions:**
Use symbolic notation and fractions where needed to compute the solution.
**Solution:**
Find \(\frac{\partial g}{\partial \theta}\) using the chain rule and express your answer in the box provided:
\[
\frac{\partial g}{\partial \theta} \bigg|_{(r, \theta)} = \boxed{}
\]
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