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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4.5 1
![## Differentiation Problem
### Task:
Differentiate the function.
Given:
\[ f(x) = \ln(2x) \]
Find:
\[ f'(x) \ = \ \boxed{\phantom{0}} \]
### Explanation:
To differentiate the function \( f(x) = \ln(2x) \), apply the chain rule. The derivative of \(\ln(u)\) is \(\frac{1}{u} \cdot \frac{du}{dx}\).
1. Let \( u = 2x \).
2. The derivative of \( u \) with respect to \( x \), \( \frac{du}{dx} = 2 \).
Using the chain rule:
\[ f'(x) = \frac{1}{2x} \cdot 2 = \frac{2}{2x} = \frac{1}{x} \]
Therefore:
\[ f'(x) = \frac{1}{x} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F34d89729-f0b7-4660-b47f-8f1bb9a32e44%2F24c0047e-05c7-4c90-8403-942c9e744cce%2Fcekk58p_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Differentiation Problem
### Task:
Differentiate the function.
Given:
\[ f(x) = \ln(2x) \]
Find:
\[ f'(x) \ = \ \boxed{\phantom{0}} \]
### Explanation:
To differentiate the function \( f(x) = \ln(2x) \), apply the chain rule. The derivative of \(\ln(u)\) is \(\frac{1}{u} \cdot \frac{du}{dx}\).
1. Let \( u = 2x \).
2. The derivative of \( u \) with respect to \( x \), \( \frac{du}{dx} = 2 \).
Using the chain rule:
\[ f'(x) = \frac{1}{2x} \cdot 2 = \frac{2}{2x} = \frac{1}{x} \]
Therefore:
\[ f'(x) = \frac{1}{x} \]
Expert Solution
Identity and rules for differentiation
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