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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![### Derivative Using the Limit Definition
#### (a) Example Problem
Given the function:
\[ f(x) = \sqrt{\frac{a}{bx}} \]
where \( a, b, x > 0 \), use the **limit definition** to find the derivative of \( f(x) \).
#### Limit Definition of the Derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]
Explanation:
To find the derivative of the function \( f(x) = \sqrt{\frac{a}{bx}} \) using the limit definition, follow these steps:
1. **Substitute \( x+h \) into the function \( f(x) \).**
\[ f(x+h) = \sqrt{\frac{a}{b(x+h)}} \]
2. **Form the difference quotient.**
\[ \frac{f(x+h) - f(x)}{h} \]
3. **Take the limit as \( h \) approaches 0.**
\[ \lim_{h \to 0} \frac{\sqrt{\frac{a}{b(x+h)}} - \sqrt{\frac{a}{bx}}}{h} \]
By evaluating this limit, you will find the derivative \( f'(x) \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2cc04b1e-803c-4d9d-b550-d2ae04bcc2b3%2Ff4fe64ee-e6d6-438b-bb68-358b3f8c7ad5%2Fm7h282p_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Derivative Using the Limit Definition
#### (a) Example Problem
Given the function:
\[ f(x) = \sqrt{\frac{a}{bx}} \]
where \( a, b, x > 0 \), use the **limit definition** to find the derivative of \( f(x) \).
#### Limit Definition of the Derivative:
\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]
Explanation:
To find the derivative of the function \( f(x) = \sqrt{\frac{a}{bx}} \) using the limit definition, follow these steps:
1. **Substitute \( x+h \) into the function \( f(x) \).**
\[ f(x+h) = \sqrt{\frac{a}{b(x+h)}} \]
2. **Form the difference quotient.**
\[ \frac{f(x+h) - f(x)}{h} \]
3. **Take the limit as \( h \) approaches 0.**
\[ \lim_{h \to 0} \frac{\sqrt{\frac{a}{b(x+h)}} - \sqrt{\frac{a}{bx}}}{h} \]
By evaluating this limit, you will find the derivative \( f'(x) \).
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