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 Statement:**
Given the equation \(x^2 + xy + y^2 = 19\).
Find \(\frac{dy}{dx}\).
**Solution Approach (Implicit Differentiation):**
To find \(\frac{dy}{dx}\) using implicit differentiation, follow these steps:
1. Differentiate both sides of the equation with respect to \(x\).
2. Apply the chain rule where necessary.
3. Solve for \(\frac{dy}{dx}\).
**Detailed Steps:**
- Differentiate \(x^2\) to get \(2x\).
- Differentiate \(xy\) using the product rule: the derivative is \(x \frac{dy}{dx} + y\).
- Differentiate \(y^2\) as \(2y \frac{dy}{dx}\).
Putting it all together:
\[
2x + (x \frac{dy}{dx} + y) + 2y \frac{dy}{dx} = 0
\]
- Rearrange to solve for \(\frac{dy}{dx}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdeff5ce5-c161-409f-a0ee-bdeaef8b0965%2F1caff34e-5dcf-4234-9779-44e9ab7d0e5b%2F08m5ezf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Given the equation \(x^2 + xy + y^2 = 19\).
Find \(\frac{dy}{dx}\).
**Solution Approach (Implicit Differentiation):**
To find \(\frac{dy}{dx}\) using implicit differentiation, follow these steps:
1. Differentiate both sides of the equation with respect to \(x\).
2. Apply the chain rule where necessary.
3. Solve for \(\frac{dy}{dx}\).
**Detailed Steps:**
- Differentiate \(x^2\) to get \(2x\).
- Differentiate \(xy\) using the product rule: the derivative is \(x \frac{dy}{dx} + y\).
- Differentiate \(y^2\) as \(2y \frac{dy}{dx}\).
Putting it all together:
\[
2x + (x \frac{dy}{dx} + y) + 2y \frac{dy}{dx} = 0
\]
- Rearrange to solve for \(\frac{dy}{dx}\).
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