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
![**Problem:**
Given the function \( y = x \sqrt{x} \), find \( \frac{dy}{dx} \).
---
**Solution:**
To find the derivative of the function \( y = x \sqrt{x} \), we will apply the product rule and the chain rule of differentiation.
1. **Rewrite the function in exponent form:**
\[
y = x \cdot x^{1/2} = x^{1 + \frac{1}{2}} = x^{3/2}
\]
2. **Differentiate using the power rule:**
The power rule states that if \( y = x^n \), then \( \frac{dy}{dx} = n x^{n-1} \).
\[
y = x^{3/2}
\]
Using the power rule:
\[
\frac{dy}{dx} = \frac{3}{2} x^{\frac{3}{2} - 1} = \frac{3}{2} x^{\frac{1}{2}} = \frac{3}{2} \sqrt{x}
\]
Thus, the derivative of \( y = x \sqrt{x} \) is:
\[
\frac{dy}{dx} = \frac{3}{2} \sqrt{x}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa2133091-a753-411c-beef-444bf7f4574e%2Fc5a34f48-7939-456e-9fe1-283165a2ca25%2Fec0f07e_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem:**
Given the function \( y = x \sqrt{x} \), find \( \frac{dy}{dx} \).
---
**Solution:**
To find the derivative of the function \( y = x \sqrt{x} \), we will apply the product rule and the chain rule of differentiation.
1. **Rewrite the function in exponent form:**
\[
y = x \cdot x^{1/2} = x^{1 + \frac{1}{2}} = x^{3/2}
\]
2. **Differentiate using the power rule:**
The power rule states that if \( y = x^n \), then \( \frac{dy}{dx} = n x^{n-1} \).
\[
y = x^{3/2}
\]
Using the power rule:
\[
\frac{dy}{dx} = \frac{3}{2} x^{\frac{3}{2} - 1} = \frac{3}{2} x^{\frac{1}{2}} = \frac{3}{2} \sqrt{x}
\]
Thus, the derivative of \( y = x \sqrt{x} \) is:
\[
\frac{dy}{dx} = \frac{3}{2} \sqrt{x}
\]
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