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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![**Differentiation Practice Problem**
**Objective:** Differentiate the following function.
Given the function:
\[ y = \ln \left( \frac{2x}{3} \right) \]
**Instructions:**
1. Use the chain rule to differentiate the natural logarithm function.
2. Apply basic differentiation rules to find the derivative \( \frac{dy}{dx} \).
3. Remember that the derivative of \( \ln(u) \) with respect to \( x \) is \( \frac{1}{u} \cdot \frac{du}{dx} \).
**Solution Steps:**
- Identify \( u = \frac{2x}{3} \).
- Differentiate \( u \) with respect to \( x \) to find \( \frac{du}{dx} \).
- Substitute \( u \) and \( \frac{du}{dx} \) into the formula for \( \frac{dy}{dx} \).
**Expected Outcome:**
The derivative \( \frac{dy}{dx} \) should exhibit how changes in \( x \) affect the logarithm of the given function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F83b86ac8-1214-407a-8ec2-632d5098b00d%2F25a7fcdd-13f7-4e78-8cc3-d9332fc68d46%2F140amn6.jpeg&w=3840&q=75)
Transcribed Image Text:**Differentiation Practice Problem**
**Objective:** Differentiate the following function.
Given the function:
\[ y = \ln \left( \frac{2x}{3} \right) \]
**Instructions:**
1. Use the chain rule to differentiate the natural logarithm function.
2. Apply basic differentiation rules to find the derivative \( \frac{dy}{dx} \).
3. Remember that the derivative of \( \ln(u) \) with respect to \( x \) is \( \frac{1}{u} \cdot \frac{du}{dx} \).
**Solution Steps:**
- Identify \( u = \frac{2x}{3} \).
- Differentiate \( u \) with respect to \( x \) to find \( \frac{du}{dx} \).
- Substitute \( u \) and \( \frac{du}{dx} \) into the formula for \( \frac{dy}{dx} \).
**Expected Outcome:**
The derivative \( \frac{dy}{dx} \) should exhibit how changes in \( x \) affect the logarithm of the given function.
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