Use the Chain Rule to find the indicated partial derivatives. z = x² + x²y₁ x = s + 2t - u, y = stu²; дz дz дz when s = 4, t = 3, u = 5 əz əs дz at əz au || = || -- as at au
Use the Chain Rule to find the indicated partial derivatives. z = x² + x²y₁ x = s + 2t - u, y = stu²; дz дz дz when s = 4, t = 3, u = 5 əz əs дz at əz au || = || -- as at au
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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![### Chain Rule for Partial Derivatives
In this exercise, we will use the Chain Rule to find the indicated partial derivatives.
The function is given as:
\[ z = x^4 + x^2y \]
with the substitutions:
\[ x = s + 2t - u \]
\[ y = stu^2 \]
We need to find the partial derivatives of \( z \) with respect to \( s \), \( t \), and \( u \) when \( s = 4 \), \( t = 3 \), and \( u = 5 \).
The partial derivatives to be evaluated are:
\[ \frac{\partial z}{\partial s}, \frac{\partial z}{\partial t}, \frac{\partial z}{\partial u} \]
### Required Steps
1. **Find the partial derivatives of \( z \) with respect to \( x \) and \( y \)**:
\[ \frac{\partial z}{\partial x} = 4x^3 + 2xy \]
\[ \frac{\partial z}{\partial y} = x^2 \]
2. **Determine the partial derivatives of \( x \) and \( y \) with respect to \( s, t, \) and \( u \)**:
\[ \frac{\partial x}{\partial s} = 1 \]
\[ \frac{\partial x}{\partial t} = 2 \]
\[ \frac{\partial x}{\partial u} = -1 \]
\[ \frac{\partial y}{\partial s} = tu^2 \]
\[ \frac{\partial y}{\partial t} = su^2 \]
\[ \frac{\partial y}{\partial u} = 2stu \]
3. **Apply the Chain Rule**:
\[ \frac{\partial z}{\partial s} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial s} \]
\[ \frac{\partial z}{\partial t} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial t} \]
\[ \frac{\partial z}{\partial u} = \frac{\partial z}{\partial](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F51323dd8-f785-4fee-9caa-f5ed8457fa8f%2Fd9bbcf79-5e04-488f-8282-fa26e200fd67%2F3nqw8bul_processed.png&w=3840&q=75)
Transcribed Image Text:### Chain Rule for Partial Derivatives
In this exercise, we will use the Chain Rule to find the indicated partial derivatives.
The function is given as:
\[ z = x^4 + x^2y \]
with the substitutions:
\[ x = s + 2t - u \]
\[ y = stu^2 \]
We need to find the partial derivatives of \( z \) with respect to \( s \), \( t \), and \( u \) when \( s = 4 \), \( t = 3 \), and \( u = 5 \).
The partial derivatives to be evaluated are:
\[ \frac{\partial z}{\partial s}, \frac{\partial z}{\partial t}, \frac{\partial z}{\partial u} \]
### Required Steps
1. **Find the partial derivatives of \( z \) with respect to \( x \) and \( y \)**:
\[ \frac{\partial z}{\partial x} = 4x^3 + 2xy \]
\[ \frac{\partial z}{\partial y} = x^2 \]
2. **Determine the partial derivatives of \( x \) and \( y \) with respect to \( s, t, \) and \( u \)**:
\[ \frac{\partial x}{\partial s} = 1 \]
\[ \frac{\partial x}{\partial t} = 2 \]
\[ \frac{\partial x}{\partial u} = -1 \]
\[ \frac{\partial y}{\partial s} = tu^2 \]
\[ \frac{\partial y}{\partial t} = su^2 \]
\[ \frac{\partial y}{\partial u} = 2stu \]
3. **Apply the Chain Rule**:
\[ \frac{\partial z}{\partial s} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial s} \]
\[ \frac{\partial z}{\partial t} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial t} \]
\[ \frac{\partial z}{\partial u} = \frac{\partial z}{\partial
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