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Find the first and second derivative of the function. G(r) = Vr + Vr 1 + 1 G'(r) = 3x G"(r) =

Find the first and second derivative of the function. G(r) = Vr + Vr 1 + 1 G'(r) = 3x G"(r) =

Calculus: Early Transcendentals
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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2.3 pt 2

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**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.
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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