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
4.1 #4
Pt3
![**Problem Statement:**
Find the derivative of the function.
\[ y = 4\sqrt{x} + 3x^{\frac{1}{8}} \]
---
**Solution:**
To find the derivative \(\frac{dy}{dx}\) of the given function, break down each term separately:
1. **Derivative of \(4\sqrt{x}\):**
Recall that \(\sqrt{x} = x^{\frac{1}{2}}\).
\[
\frac{d}{dx}[4x^{\frac{1}{2}}] = 4 \cdot \frac{1}{2}x^{\frac{1}{2}-1} = 2x^{-\frac{1}{2}}
\]
Simplifying further:
\[
2x^{-\frac{1}{2}} = \frac{2}{\sqrt{x}}
\]
2. **Derivative of \(3x^{\frac{1}{8}}\):**
\[
\frac{d}{dx}[3x^{\frac{1}{8}}] = 3 \cdot \frac{1}{8}x^{\frac{1}{8}-1} = \frac{3}{8}x^{-\frac{7}{8}}
\]
By combining the derivatives of both terms, we have:
\[
\frac{dy}{dx} = \frac{2}{\sqrt{x}} + \frac{3}{8}x^{-\frac{7}{8}}
\]
The box for \(\frac{dy}{dx}\) is left for further input or checking the solution as needed.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F63591bc8-75d6-42dc-963a-0ef4cf7b290a%2F94cd29c7-4a91-4555-86e9-773d55ae1783%2Fj8r307v_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the derivative of the function.
\[ y = 4\sqrt{x} + 3x^{\frac{1}{8}} \]
---
**Solution:**
To find the derivative \(\frac{dy}{dx}\) of the given function, break down each term separately:
1. **Derivative of \(4\sqrt{x}\):**
Recall that \(\sqrt{x} = x^{\frac{1}{2}}\).
\[
\frac{d}{dx}[4x^{\frac{1}{2}}] = 4 \cdot \frac{1}{2}x^{\frac{1}{2}-1} = 2x^{-\frac{1}{2}}
\]
Simplifying further:
\[
2x^{-\frac{1}{2}} = \frac{2}{\sqrt{x}}
\]
2. **Derivative of \(3x^{\frac{1}{8}}\):**
\[
\frac{d}{dx}[3x^{\frac{1}{8}}] = 3 \cdot \frac{1}{8}x^{\frac{1}{8}-1} = \frac{3}{8}x^{-\frac{7}{8}}
\]
By combining the derivatives of both terms, we have:
\[
\frac{dy}{dx} = \frac{2}{\sqrt{x}} + \frac{3}{8}x^{-\frac{7}{8}}
\]
The box for \(\frac{dy}{dx}\) is left for further input or checking the solution as needed.
Expert Solution
Step 1
Use formulas of derivatives to find dy/dx
Step by step
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