Here are exercises that focus on finding the partial derivatives of given functions with respect to \( x \) and \( y \). 6. For the function \( f(x, y) = \ln(1 + x^2y^2) \): a. Find the partial derivative with respect to \( x \), denoted as \( f_x(x, y) \). b. Find the partial derivative with respect to \( y \), denoted as \( f_y(x, y) \). 7. For the function \( f(x, y) = e^{x^2 + y^2} \): a. Find the partial derivative with respect to \( x \), denoted as \( f_x(x, y) \). b. Find the partial derivative with respect to \( y \), denoted as \( f_y(x, y) \). 8. For the function \( f(x, y) = (x^3 + x^2y + 3y^2)^3 \): a. Find the partial derivative with respect to \( x \), denoted as \( f_x(x, y) \). b. Find the partial derivative with respect to \( y \), denoted as \( f_y(x, y) \). These exercises will strengthen your understanding of partial differentiation, a fundamental concept in multivariable calculus.

Here are exercises that focus on finding the partial derivatives of given functions with respect to \( x \) and \( y \). 6. For the function \( f(x, y) = \ln(1 + x^2y^2) \): a. Find the partial derivative with respect to \( x \), denoted as \( f_x(x, y) \). b. Find the partial derivative with respect to \( y \), denoted as \( f_y(x, y) \). 7. For the function \( f(x, y) = e^{x^2 + y^2} \): a. Find the partial derivative with respect to \( x \), denoted as \( f_x(x, y) \). b. Find the partial derivative with respect to \( y \), denoted as \( f_y(x, y) \). 8. For the function \( f(x, y) = (x^3 + x^2y + 3y^2)^3 \): a. Find the partial derivative with respect to \( x \), denoted as \( f_x(x, y) \). b. Find the partial derivative with respect to \( y \), denoted as \( f_y(x, y) \). These exercises will strengthen your understanding of partial differentiation, a fundamental concept in multivariable calculus.

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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Here are exercises that focus on finding the partial derivatives of given functions with respect to \( x \) and \( y \).

6. For the function \( f(x, y) = \ln(1 + x^2y^2) \):

a. Find the partial derivative with respect to \( x \), denoted as \( f_x(x, y) \).

b. Find the partial derivative with respect to \( y \), denoted as \( f_y(x, y) \).

7. For the function \( f(x, y) = e^{x^2 + y^2} \):

a. Find the partial derivative with respect to \( x \), denoted as \( f_x(x, y) \).

b. Find the partial derivative with respect to \( y \), denoted as \( f_y(x, y) \).

8. For the function \( f(x, y) = (x^3 + x^2y + 3y^2)^3 \):

a. Find the partial derivative with respect to \( x \), denoted as \( f_x(x, y) \).

b. Find the partial derivative with respect to \( y \), denoted as \( f_y(x, y) \).

These exercises will strengthen your understanding of partial differentiation, a fundamental concept in multivariable calculus.

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