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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![**Educational Explanation of Partial Derivatives**
**Problem Statement:**
Find the first partial derivatives of the function:
\[
f(x, y) = \frac{4x - y}{4x + y}
\]
at the point \((x, y) = (2, 3)\).
**Tasks:**
1. Calculate the partial derivative of \(f\) with respect to \(x\) at the point \((2, 3)\):
\[
\frac{\partial f}{\partial x}(2, 3) = \underline{\hspace{3cm}}
\]
2. Calculate the partial derivative of \(f\) with respect to \(y\) at the point \((2, 3)\):
\[
\frac{\partial f}{\partial y}(2, 3) = \underline{\hspace{3cm}}
\]
---
**Understanding Partial Derivatives:**
Partial derivatives represent how a multivariable function changes as one of the variables changes, holding the other variables constant.
1. **Partial Derivative with respect to \(x\)**: This tells us the rate at which \(f(x, y)\) changes as \(x\) changes, while \(y\) is held constant.
2. **Partial Derivative with respect to \(y\)**: This tells us the rate at which \(f(x, y)\) changes as \(y\) changes, while \(x\) is held constant.
To solve the problem, we can apply the rules of differentiation to find each partial derivative, then evaluate them at the given point \((2, 3)\).
This involves using techniques such as the quotient rule for differentiation, as the function \(f(x, y)\) is a ratio of two expressions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7cce1808-caf5-457b-bc5a-f7c079ff1a3b%2F8027a27d-10f7-4741-8fd0-478987bf36cc%2Ftkyvchj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Educational Explanation of Partial Derivatives**
**Problem Statement:**
Find the first partial derivatives of the function:
\[
f(x, y) = \frac{4x - y}{4x + y}
\]
at the point \((x, y) = (2, 3)\).
**Tasks:**
1. Calculate the partial derivative of \(f\) with respect to \(x\) at the point \((2, 3)\):
\[
\frac{\partial f}{\partial x}(2, 3) = \underline{\hspace{3cm}}
\]
2. Calculate the partial derivative of \(f\) with respect to \(y\) at the point \((2, 3)\):
\[
\frac{\partial f}{\partial y}(2, 3) = \underline{\hspace{3cm}}
\]
---
**Understanding Partial Derivatives:**
Partial derivatives represent how a multivariable function changes as one of the variables changes, holding the other variables constant.
1. **Partial Derivative with respect to \(x\)**: This tells us the rate at which \(f(x, y)\) changes as \(x\) changes, while \(y\) is held constant.
2. **Partial Derivative with respect to \(y\)**: This tells us the rate at which \(f(x, y)\) changes as \(y\) changes, while \(x\) is held constant.
To solve the problem, we can apply the rules of differentiation to find each partial derivative, then evaluate them at the given point \((2, 3)\).
This involves using techniques such as the quotient rule for differentiation, as the function \(f(x, y)\) is a ratio of two expressions.
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