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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Question
![**Topic: Partial Derivatives in Multivariable Calculus**
**Objective:**
Find \(\frac{\partial f}{\partial x}\) for \(z = f(1, 1)\), given the function:
**Function Definition:**
\[
z = f(x, y) = 4xy + z^3 x + 2yz = 4
\]
**Explanation:**
This problem requires finding the partial derivative of a multivariable function. The function \(z\) is expressed in terms of \(x\) and \(y\). The objective is to determine how \(z\) changes with respect to \(x\) while keeping \(y\) constant. The expression also includes a term \(z^3 x\), which implies the dependency of \(z\) on both \(x\) and \(y\), indicating a potentially implicit differentiation scenario.
**Steps:**
1. Understand that implicit differentiation might be necessary due to the presence of \(z\) on both sides.
2. Find \(\frac{\partial f}{\partial x}\) by applying differentiation rules for each term.
3. Evaluate the derivative at \(z = f(1, 1)\) after solving the equation for \(z\) if needed.
This exercise enhances understanding of partial derivatives and implicit differentiation by exploring how a multivariable function's output changes concerning one input variable while others remain fixed.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe620d41c-dd63-496b-b84c-cb17355be0e7%2F0a4ebb52-dce6-49ac-826a-e98402898990%2Fxds9gyo.jpeg&w=3840&q=75)
Transcribed Image Text:**Topic: Partial Derivatives in Multivariable Calculus**
**Objective:**
Find \(\frac{\partial f}{\partial x}\) for \(z = f(1, 1)\), given the function:
**Function Definition:**
\[
z = f(x, y) = 4xy + z^3 x + 2yz = 4
\]
**Explanation:**
This problem requires finding the partial derivative of a multivariable function. The function \(z\) is expressed in terms of \(x\) and \(y\). The objective is to determine how \(z\) changes with respect to \(x\) while keeping \(y\) constant. The expression also includes a term \(z^3 x\), which implies the dependency of \(z\) on both \(x\) and \(y\), indicating a potentially implicit differentiation scenario.
**Steps:**
1. Understand that implicit differentiation might be necessary due to the presence of \(z\) on both sides.
2. Find \(\frac{\partial f}{\partial x}\) by applying differentiation rules for each term.
3. Evaluate the derivative at \(z = f(1, 1)\) after solving the equation for \(z\) if needed.
This exercise enhances understanding of partial derivatives and implicit differentiation by exploring how a multivariable function's output changes concerning one input variable while others remain fixed.
Expert Solution
Step 1
Derivative: - The function's derivative is the equation we discovered. The derivative is defined as the instantaneous rate of change, or slope, at a certain point in a function, as described previously. It tells you the exact slope of the curve at a specified location. denotes the derivative of with respect to.
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