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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![### Problem Statement
If \( f(x) = \frac{\sqrt{x} - 4}{\sqrt{x} + 4} \), find:
\( f'(x) \) = ________
\( f'(2) \) = ________
### Instructions
To solve this problem, you will need to:
1. Differentiate the given function \( f(x) \) with respect to \( x \) to find \( f'(x) \).
2. Evaluate the derivative at \( x = 2 \) to find \( f'(2) \).
### Solution Steps
1. **Differentiate \( f(x) \) with respect to \( x \).**
2. **Evaluate the derivative at \( x = 2 \).**
### Additional Information
You can use the quotient rule for differentiation since \( f(x) \) is given in the form of a quotient of two functions. The quotient rule is given by:
\[ \left( \frac{u}{v} \right)' = \frac{v \cdot u' - u \cdot v'}{v^2} \]
where \( u = \sqrt{x} - 4 \) and \( v = \sqrt{x} + 4 \).
Fill in the answers in the provided boxes after performing the necessary calculations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6f6b4032-4723-40f9-bafe-2ad74429fe12%2Fba9b01b4-6937-430f-ba28-fe455eb24661%2Fqm9ef69_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
If \( f(x) = \frac{\sqrt{x} - 4}{\sqrt{x} + 4} \), find:
\( f'(x) \) = ________
\( f'(2) \) = ________
### Instructions
To solve this problem, you will need to:
1. Differentiate the given function \( f(x) \) with respect to \( x \) to find \( f'(x) \).
2. Evaluate the derivative at \( x = 2 \) to find \( f'(2) \).
### Solution Steps
1. **Differentiate \( f(x) \) with respect to \( x \).**
2. **Evaluate the derivative at \( x = 2 \).**
### Additional Information
You can use the quotient rule for differentiation since \( f(x) \) is given in the form of a quotient of two functions. The quotient rule is given by:
\[ \left( \frac{u}{v} \right)' = \frac{v \cdot u' - u \cdot v'}{v^2} \]
where \( u = \sqrt{x} - 4 \) and \( v = \sqrt{x} + 4 \).
Fill in the answers in the provided boxes after performing the necessary calculations.
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