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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![**Finding \( \frac{dy}{dx} \) by Implicit Differentiation**
**Problem Statement:**
Find \( \frac{dy}{dx} \) by implicit differentiation for the following equation:
\[ (5x + 2y)^{1/3} = x^2 \]
**Solution:**
To find \( \frac{dy}{dx} \) by implicit differentiation, follow these steps:
1. Differentiate both sides of the equation with respect to \( x \).
\[ \frac{d}{dx} \left( (5x + 2y)^{1/3} \right) = \frac{d}{dx} \left( x^2 \right) \]
2. Use the chain rule on the left side. The derivative of \( (5x + 2y)^{1/3} \) with respect to \( x \) is:
\[ \frac{1}{3} (5x + 2y)^{-2/3} \cdot \frac{d}{dx} (5x + 2y) \]
3. Next, differentiate the inner function \( 5x + 2y \) with respect to \( x \). Remember that \( y \) is a function of \( x \):
\[ \frac{d}{dx} (5x + 2y) = 5 + 2 \frac{dy}{dx} \]
4. Therefore, the left side becomes:
\[ \frac{1}{3} (5x + 2y)^{-2/3} (5 + 2 \frac{dy}{dx}) \]
5. Differentiate the right side:
\[ \frac{d}{dx} (x^2) = 2x \]
6. Set the derivatives equal to each other:
\[ \frac{1}{3} (5x + 2y)^{-2/3} (5 + 2 \frac{dy}{dx}) = 2x \]
7. Solve for \( \frac{dy}{dx} \):
\[ 5 + 2 \frac{dy}{dx} = 6x (5x + 2y)^{2/3} \]
\[ 2 \frac{dy}{dx} = 6x (5x +](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F885800f4-ea55-4813-8161-e1df31cc31af%2F3a949ef6-cb99-4fea-be72-f043015495dd%2F8brsl5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Finding \( \frac{dy}{dx} \) by Implicit Differentiation**
**Problem Statement:**
Find \( \frac{dy}{dx} \) by implicit differentiation for the following equation:
\[ (5x + 2y)^{1/3} = x^2 \]
**Solution:**
To find \( \frac{dy}{dx} \) by implicit differentiation, follow these steps:
1. Differentiate both sides of the equation with respect to \( x \).
\[ \frac{d}{dx} \left( (5x + 2y)^{1/3} \right) = \frac{d}{dx} \left( x^2 \right) \]
2. Use the chain rule on the left side. The derivative of \( (5x + 2y)^{1/3} \) with respect to \( x \) is:
\[ \frac{1}{3} (5x + 2y)^{-2/3} \cdot \frac{d}{dx} (5x + 2y) \]
3. Next, differentiate the inner function \( 5x + 2y \) with respect to \( x \). Remember that \( y \) is a function of \( x \):
\[ \frac{d}{dx} (5x + 2y) = 5 + 2 \frac{dy}{dx} \]
4. Therefore, the left side becomes:
\[ \frac{1}{3} (5x + 2y)^{-2/3} (5 + 2 \frac{dy}{dx}) \]
5. Differentiate the right side:
\[ \frac{d}{dx} (x^2) = 2x \]
6. Set the derivatives equal to each other:
\[ \frac{1}{3} (5x + 2y)^{-2/3} (5 + 2 \frac{dy}{dx}) = 2x \]
7. Solve for \( \frac{dy}{dx} \):
\[ 5 + 2 \frac{dy}{dx} = 6x (5x + 2y)^{2/3} \]
\[ 2 \frac{dy}{dx} = 6x (5x +
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