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
![**Differentiation Problem**
The task is to find the derivative of the function defined as follows:
\[ f(x) = \sqrt{x} + \sqrt{x} \]
### Solution Approach
To differentiate the function \( f(x) \), we apply the basic differentiation rules.
The function can be rewritten as:
\[ f(x) = 2\sqrt{x} \]
Next, apply the power rule for differentiation. Recall that the derivative of \( x^n \) is \( nx^{n-1} \).
1. Rewrite \( \sqrt{x} \) as \( x^{1/2} \).
2. Differentiate \( 2x^{1/2} \) using the power rule.
#### Steps:
- The derivative of \( x^{1/2} \) is \( \frac{1}{2}x^{-1/2} \).
- Multiply by the constant 2.
Thus, the derivative \( f'(x) \) will be:
\[ f'(x) = 2 \cdot \frac{1}{2}x^{-1/2} \]
Simplifying, we get:
\[ f'(x) = x^{-1/2} \]
Therefore, in terms of radicals:
\[ f'(x) = \frac{1}{\sqrt{x}} \]
This completes the differentiation process for the given function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faeadb183-1175-48ee-be1d-be4cbd36e1e9%2Fb320419a-cb18-43f8-9f33-8c8ab764b3ad%2Fasq2n0s_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Differentiation Problem**
The task is to find the derivative of the function defined as follows:
\[ f(x) = \sqrt{x} + \sqrt{x} \]
### Solution Approach
To differentiate the function \( f(x) \), we apply the basic differentiation rules.
The function can be rewritten as:
\[ f(x) = 2\sqrt{x} \]
Next, apply the power rule for differentiation. Recall that the derivative of \( x^n \) is \( nx^{n-1} \).
1. Rewrite \( \sqrt{x} \) as \( x^{1/2} \).
2. Differentiate \( 2x^{1/2} \) using the power rule.
#### Steps:
- The derivative of \( x^{1/2} \) is \( \frac{1}{2}x^{-1/2} \).
- Multiply by the constant 2.
Thus, the derivative \( f'(x) \) will be:
\[ f'(x) = 2 \cdot \frac{1}{2}x^{-1/2} \]
Simplifying, we get:
\[ f'(x) = x^{-1/2} \]
Therefore, in terms of radicals:
\[ f'(x) = \frac{1}{\sqrt{x}} \]
This completes the differentiation process for the given function.
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