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...
Related questions
Question
2.3 pt 2
12
![**Problem Statement:**
Find the first and second derivative of the function:
\[ G(r) = \sqrt{r} + \sqrt[3]{r} \]
**Solution Steps:**
1. **First Derivative Attempt:**
\[
G'(r) = \frac{1}{2\sqrt{x}} + \frac{1}{3x^{\frac{2}{3}}} \quad \text{(Incorrect Solution, indicated by a red 'X')}
\]
2. **Second Derivative:**
\[
G''(r) = \quad \text{(Box for input)}
\]
**Enhanced Feedback:**
Remember to use the Power Rule:
\[
\frac{d}{dx} (x^n) = nx^{n-1}
\]
Keep in mind that \(\sqrt[n]{r} = r^{1/n}\) and \(\sqrt{r} = r^{1/2}\). The second derivative of a function is the derivative of the derivative of that function. Therefore, first find the first derivative, then take the derivative of the first derivative to find the second derivative.
**Additional Help:**
- If you need further assistance, you can:
- **Read It**: Click the button for a detailed explanation.
- **Watch It**: Click the button for a video tutorial.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3241d4b2-7cea-4408-ac37-56f0a05274af%2F2975fea7-ac2a-41bf-ba6b-b62b7ee36118%2F1ehad7e_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the first and second derivative of the function:
\[ G(r) = \sqrt{r} + \sqrt[3]{r} \]
**Solution Steps:**
1. **First Derivative Attempt:**
\[
G'(r) = \frac{1}{2\sqrt{x}} + \frac{1}{3x^{\frac{2}{3}}} \quad \text{(Incorrect Solution, indicated by a red 'X')}
\]
2. **Second Derivative:**
\[
G''(r) = \quad \text{(Box for input)}
\]
**Enhanced Feedback:**
Remember to use the Power Rule:
\[
\frac{d}{dx} (x^n) = nx^{n-1}
\]
Keep in mind that \(\sqrt[n]{r} = r^{1/n}\) and \(\sqrt{r} = r^{1/2}\). The second derivative of a function is the derivative of the derivative of that function. Therefore, first find the first derivative, then take the derivative of the first derivative to find the second derivative.
**Additional Help:**
- If you need further assistance, you can:
- **Read It**: Click the button for a detailed explanation.
- **Watch It**: Click the button for a video tutorial.
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