**Problem 45:** Find the partial derivatives \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\) at the point \((-1, 1, -1)\) for the given implicit function: \[ y e^{y-x^2} + z^2 = -3z - y^3 \] ### Solution Summary: This problem requires the application of implicit differentiation to find the partial derivatives \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\) for the given function at the specified point. The procedure typically involves differentiating both sides of the equation with respect to \(x\) and \(y\) separately, treating \(z\) as a function of \(x\) and \(y\), and then solving the resulting system of equations. ### Steps to Solve: 1. Differentiate the equation implicitly with respect to \(x\) to find \(\frac{\partial z}{\partial x}\). 2. Differentiate the equation implicitly with respect to \(y\) to find \(\frac{\partial z}{\partial y}\). 3. Substitute the point \((-1, 1, -1)\) into the resulting expressions to find the numerical values of the partial derivatives.
**Problem 45:** Find the partial derivatives \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\) at the point \((-1, 1, -1)\) for the given implicit function: \[ y e^{y-x^2} + z^2 = -3z - y^3 \] ### Solution Summary: This problem requires the application of implicit differentiation to find the partial derivatives \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\) for the given function at the specified point. The procedure typically involves differentiating both sides of the equation with respect to \(x\) and \(y\) separately, treating \(z\) as a function of \(x\) and \(y\), and then solving the resulting system of equations. ### Steps to Solve: 1. Differentiate the equation implicitly with respect to \(x\) to find \(\frac{\partial z}{\partial x}\). 2. Differentiate the equation implicitly with respect to \(y\) to find \(\frac{\partial z}{\partial y}\). 3. Substitute the point \((-1, 1, -1)\) into the resulting expressions to find the numerical values of the partial derivatives.
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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![**Problem 45:**
Find the partial derivatives \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\) at the point \((-1, 1, -1)\) for the given implicit function:
\[ y e^{y-x^2} + z^2 = -3z - y^3 \]
### Solution Summary:
This problem requires the application of implicit differentiation to find the partial derivatives \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\) for the given function at the specified point. The procedure typically involves differentiating both sides of the equation with respect to \(x\) and \(y\) separately, treating \(z\) as a function of \(x\) and \(y\), and then solving the resulting system of equations.
### Steps to Solve:
1. Differentiate the equation implicitly with respect to \(x\) to find \(\frac{\partial z}{\partial x}\).
2. Differentiate the equation implicitly with respect to \(y\) to find \(\frac{\partial z}{\partial y}\).
3. Substitute the point \((-1, 1, -1)\) into the resulting expressions to find the numerical values of the partial derivatives.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F18107b7c-8d8e-412b-a2cd-6addf4186be2%2F8d87a071-1c45-4e99-8f66-3711a68b0375%2Fj6ay8e6_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 45:**
Find the partial derivatives \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\) at the point \((-1, 1, -1)\) for the given implicit function:
\[ y e^{y-x^2} + z^2 = -3z - y^3 \]
### Solution Summary:
This problem requires the application of implicit differentiation to find the partial derivatives \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\) for the given function at the specified point. The procedure typically involves differentiating both sides of the equation with respect to \(x\) and \(y\) separately, treating \(z\) as a function of \(x\) and \(y\), and then solving the resulting system of equations.
### Steps to Solve:
1. Differentiate the equation implicitly with respect to \(x\) to find \(\frac{\partial z}{\partial x}\).
2. Differentiate the equation implicitly with respect to \(y\) to find \(\frac{\partial z}{\partial y}\).
3. Substitute the point \((-1, 1, -1)\) into the resulting expressions to find the numerical values of the partial derivatives.
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