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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6.4 #2
![### Implicit Differentiation Exercise
**Problem Statement:**
Find \(\frac{dy}{dx}\) using implicit differentiation for the equation:
\[ 2x^3 = 5y^2 + 7y \]
**Solution Steps:**
To solve this problem, you need to differentiate both sides of the equation with respect to \(x\), applying the rules for implicit differentiation. Remember that when differentiating terms involving \(y\), you also multiply by \(\frac{dy}{dx}\), since \(y\) is considered a function of \(x\).
1. Differentiate both sides of the equation with respect to \(x\):
- For the left side: \(\frac{d}{dx}(2x^3) = 6x^2\).
- For the right side: differentiate each term \(\frac{d}{dx}(5y^2 + 7y)\):
- \(5y^2\) becomes \(10y \cdot \frac{dy}{dx}\)
- \(7y\) becomes \(7 \cdot \frac{dy}{dx}\)
2. Combine the derived terms:
\[
6x^2 = 10y \cdot \frac{dy}{dx} + 7 \cdot \frac{dy}{dx}
\]
3. Factor out \(\frac{dy}{dx}\) on the right side:
\[
6x^2 = (10y + 7) \cdot \frac{dy}{dx}
\]
4. Solve for \(\frac{dy}{dx}\):
\[
\frac{dy}{dx} = \frac{6x^2}{10y + 7}
\]
The solution to the implicit differentiation problem is \(\frac{dy}{dx} = \frac{6x^2}{10y + 7}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F209d6d6c-b2bc-44f2-b868-e3db159d044a%2F0cd3aac8-a0c8-4ce3-9412-763036b6e092%2Fjqlh7qda_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Implicit Differentiation Exercise
**Problem Statement:**
Find \(\frac{dy}{dx}\) using implicit differentiation for the equation:
\[ 2x^3 = 5y^2 + 7y \]
**Solution Steps:**
To solve this problem, you need to differentiate both sides of the equation with respect to \(x\), applying the rules for implicit differentiation. Remember that when differentiating terms involving \(y\), you also multiply by \(\frac{dy}{dx}\), since \(y\) is considered a function of \(x\).
1. Differentiate both sides of the equation with respect to \(x\):
- For the left side: \(\frac{d}{dx}(2x^3) = 6x^2\).
- For the right side: differentiate each term \(\frac{d}{dx}(5y^2 + 7y)\):
- \(5y^2\) becomes \(10y \cdot \frac{dy}{dx}\)
- \(7y\) becomes \(7 \cdot \frac{dy}{dx}\)
2. Combine the derived terms:
\[
6x^2 = 10y \cdot \frac{dy}{dx} + 7 \cdot \frac{dy}{dx}
\]
3. Factor out \(\frac{dy}{dx}\) on the right side:
\[
6x^2 = (10y + 7) \cdot \frac{dy}{dx}
\]
4. Solve for \(\frac{dy}{dx}\):
\[
\frac{dy}{dx} = \frac{6x^2}{10y + 7}
\]
The solution to the implicit differentiation problem is \(\frac{dy}{dx} = \frac{6x^2}{10y + 7}\).
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