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
![**Problem Statement**
Find \(\frac{d^2y}{dx^2}\).
Given the equation:
\[ e^{4y} + x = 7y \]
---
**Solution**
To solve for \(\frac{d^2y}{dx^2}\), follow these steps:
1. **Differentiate the given equation implicitly with respect to \(x\).**
Start with:
\[ e^{4y} + x = 7y \]
Differentiate both sides with respect to \(x\):
\[ 4e^{4y} \frac{dy}{dx} + 1 = 7 \frac{dy}{dx} \]
2. **Solve for the first derivative \(\frac{dy}{dx}\).**
3. **Differentiate again to find \(\frac{d^2y}{dx^2}\).**
Continue from the expression for \(\frac{dy}{dx}\) and apply the derivative again.
Substitute to get \(\frac{d^2y}{dx^2}\) as required. Fill in the blank with your solution.
\[ \frac{d^2y}{dx^2} = \boxed{} \]
---
This problem involves applying implicit differentiation and solving a system of equations to find the second derivative.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F39190787-1e8c-4e2c-8802-b30850ac5f44%2F14503520-d726-402f-bf09-40ae75c1c00a%2Fs8wu9y_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
Find \(\frac{d^2y}{dx^2}\).
Given the equation:
\[ e^{4y} + x = 7y \]
---
**Solution**
To solve for \(\frac{d^2y}{dx^2}\), follow these steps:
1. **Differentiate the given equation implicitly with respect to \(x\).**
Start with:
\[ e^{4y} + x = 7y \]
Differentiate both sides with respect to \(x\):
\[ 4e^{4y} \frac{dy}{dx} + 1 = 7 \frac{dy}{dx} \]
2. **Solve for the first derivative \(\frac{dy}{dx}\).**
3. **Differentiate again to find \(\frac{d^2y}{dx^2}\).**
Continue from the expression for \(\frac{dy}{dx}\) and apply the derivative again.
Substitute to get \(\frac{d^2y}{dx^2}\) as required. Fill in the blank with your solution.
\[ \frac{d^2y}{dx^2} = \boxed{} \]
---
This problem involves applying implicit differentiation and solving a system of equations to find the second derivative.
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