Find ܐܐܝ ܐܝܐ ܩܐܐ DESE 4y +x=7y

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**

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.
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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