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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![Title: Understanding Second Derivatives
### Problem Statement
Find \(\frac{d^2 y}{dx^2}\) in terms of \(x\) and \(y\).
### Given Equation
\[ y^5 = x^6 \]
### Solution
\[ \frac{d^2 y}{dx^2} = \boxed{\frac{6}{25} \cdot x^{\frac{3}{5}}} \]
---
### Explanation:
In this problem, we start with the given equation \( y^5 = x^6 \). We are asked to find the second derivative of \( y \) with respect to \( x \), denoted as \(\frac{d^2 y}{dx^2}\).
1. **Implicit Differentiation**:
- First, implicitly differentiate both sides of the equation \( y^5 = x^6 \) with respect to \( x \) to find the first derivative \(\frac{dy}{dx}\).
2. **Second Derivative**:
- Next, differentiate the first derivative to find the second derivative, \(\frac{d^2 y}{dx^2}\), expressing it in terms of \( x \) and \( y \).
The detailed calculation steps may include using the chain rule and power rule for differentiation along with solving for the necessary terms step-by-step.
### Additional Notes:
- The solution box clearly indicates the result of the second derivative.
- The red 'X' at the bottom suggests an error or that the initial boxed answer is not correct.
Thus, students must review the differentiation process to ensure accuracy or consider if any steps or rules were misapplied.
This exercise highlights the importance of correctly applying calculus rules and understanding implicit differentiation to solve higher-order derivatives.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0711daf6-3b19-448b-99c0-3e7fe08f4fe3%2Fc6ff081e-8912-44f9-8de6-426b74273b3b%2Fyi07jy_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Title: Understanding Second Derivatives
### Problem Statement
Find \(\frac{d^2 y}{dx^2}\) in terms of \(x\) and \(y\).
### Given Equation
\[ y^5 = x^6 \]
### Solution
\[ \frac{d^2 y}{dx^2} = \boxed{\frac{6}{25} \cdot x^{\frac{3}{5}}} \]
---
### Explanation:
In this problem, we start with the given equation \( y^5 = x^6 \). We are asked to find the second derivative of \( y \) with respect to \( x \), denoted as \(\frac{d^2 y}{dx^2}\).
1. **Implicit Differentiation**:
- First, implicitly differentiate both sides of the equation \( y^5 = x^6 \) with respect to \( x \) to find the first derivative \(\frac{dy}{dx}\).
2. **Second Derivative**:
- Next, differentiate the first derivative to find the second derivative, \(\frac{d^2 y}{dx^2}\), expressing it in terms of \( x \) and \( y \).
The detailed calculation steps may include using the chain rule and power rule for differentiation along with solving for the necessary terms step-by-step.
### Additional Notes:
- The solution box clearly indicates the result of the second derivative.
- The red 'X' at the bottom suggests an error or that the initial boxed answer is not correct.
Thus, students must review the differentiation process to ensure accuracy or consider if any steps or rules were misapplied.
This exercise highlights the importance of correctly applying calculus rules and understanding implicit differentiation to solve higher-order derivatives.
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