Let f(x, y, z) = 8xy - z², x = Use the Chain Rule to calculate the partial derivative. (Use symbolic notation and fractions where needed. Express the answer in terms of independent variables.) af де 6r cos(0), y = :cos²(0), z = 7r. JI
Let f(x, y, z) = 8xy - z², x = Use the Chain Rule to calculate the partial derivative. (Use symbolic notation and fractions where needed. Express the answer in terms of independent variables.) af де 6r cos(0), y = :cos²(0), z = 7r. JI
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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![**Chain Rule for Partial Derivatives Example**
Given:
\[ f(x, y, z) = 8xy - z^2, \]
\[ x = 6r \cos(\theta), \]
\[ y = \cos^2(\theta), \]
\[ z = 7r. \]
**Problem:**
Use the Chain Rule to calculate the partial derivative.
**Instructions:**
(Use symbolic notation and fractions where needed. Express the answer in terms of independent variables.)
\[ \frac{\partial f}{\partial \theta} = \]
**Solution:**
To find \(\frac{\partial f}{\partial \theta}\), we will use the chain rule for multivariable functions. The chain rule states that if \(f\) is a function of \(x\), \(y\), and \(z\), and these are each functions of \(\theta\) and \(r\), we have:
\[
\frac{\partial f}{\partial \theta} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial \theta} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial \theta} + \frac{\partial f}{\partial z} \frac{\partial z}{\partial \theta}.
\]
First, we need to compute each partial derivative:
1. Calculate \(\frac{\partial f}{\partial x}\):
\[
\frac{\partial f}{\partial x} = \frac{\partial}{\partial x} (8xy - z^2) = 8y.
\]
2. Calculate \(\frac{\partial f}{\partial y}\):
\[
\frac{\partial f}{\partial y} = \frac{\partial}{\partial y} (8xy - z^2) = 8x.
\]
3. Calculate \(\frac{\partial f}{\partial z}\):
\[
\frac{\partial f}{\partial z} = \frac{\partial}{\partial z} (8xy - z^2) = -2z.
\]
Next, we need to calculate the partial derivatives of \(x\), \(y\), and \(z\) with respect to \(\theta\):
4. Calculate \(\frac{\partial x}{\partial \theta}\):
\[
x = 6r \cos(\theta),
\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F21d63f30-e516-41f1-b813-09fba408cac6%2F05bac722-df7c-48d7-b53b-d976890884fc%2Fyvsi5bh_processed.png&w=3840&q=75)
Transcribed Image Text:**Chain Rule for Partial Derivatives Example**
Given:
\[ f(x, y, z) = 8xy - z^2, \]
\[ x = 6r \cos(\theta), \]
\[ y = \cos^2(\theta), \]
\[ z = 7r. \]
**Problem:**
Use the Chain Rule to calculate the partial derivative.
**Instructions:**
(Use symbolic notation and fractions where needed. Express the answer in terms of independent variables.)
\[ \frac{\partial f}{\partial \theta} = \]
**Solution:**
To find \(\frac{\partial f}{\partial \theta}\), we will use the chain rule for multivariable functions. The chain rule states that if \(f\) is a function of \(x\), \(y\), and \(z\), and these are each functions of \(\theta\) and \(r\), we have:
\[
\frac{\partial f}{\partial \theta} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial \theta} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial \theta} + \frac{\partial f}{\partial z} \frac{\partial z}{\partial \theta}.
\]
First, we need to compute each partial derivative:
1. Calculate \(\frac{\partial f}{\partial x}\):
\[
\frac{\partial f}{\partial x} = \frac{\partial}{\partial x} (8xy - z^2) = 8y.
\]
2. Calculate \(\frac{\partial f}{\partial y}\):
\[
\frac{\partial f}{\partial y} = \frac{\partial}{\partial y} (8xy - z^2) = 8x.
\]
3. Calculate \(\frac{\partial f}{\partial z}\):
\[
\frac{\partial f}{\partial z} = \frac{\partial}{\partial z} (8xy - z^2) = -2z.
\]
Next, we need to calculate the partial derivatives of \(x\), \(y\), and \(z\) with respect to \(\theta\):
4. Calculate \(\frac{\partial x}{\partial \theta}\):
\[
x = 6r \cos(\theta),
\
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