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
(9) Please show work as necessary.
![**Topic: Calculus - Differentiation**
**Objective: Differentiate the given function.**
The function to differentiate is:
\[ f(x) = \ln\left(\frac{x^6}{8}\right) \]
---
**Solution**
To differentiate, apply the chain rule and logarithmic properties. Start by using the property of logarithms:
\[ \ln\left(\frac{x^6}{8}\right) = \ln(x^6) - \ln(8) \]
Now, differentiate each term:
1. Differentiate \(\ln(x^6)\):
- Use the chain rule: \((\ln(u))' = \frac{1}{u} \cdot u'\),
- Here \(u = x^6\) so \(u' = 6x^5\),
- Thus, \(\frac{d}{dx}[\ln(x^6)] = \frac{1}{x^6} \cdot 6x^5 = \frac{6}{x}\).
2. \(\ln(8)\) is a constant, so its derivative is 0.
The derivative of the function is then:
\[ f'(x) = \frac{6}{x} \]
---
**Result**
\[ f'(x) = \boxed{\frac{6}{x}} \]
Use this result to understand how logarithmic differentiation works and the application of different rules like the chain rule in calculus.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9dc72e72-cadd-4409-82bd-d52418d83748%2F0e1afc77-b415-4ddb-b93c-ff9e0094e0d7%2Fjxyrso_processed.png&w=3840&q=75)
Transcribed Image Text:**Topic: Calculus - Differentiation**
**Objective: Differentiate the given function.**
The function to differentiate is:
\[ f(x) = \ln\left(\frac{x^6}{8}\right) \]
---
**Solution**
To differentiate, apply the chain rule and logarithmic properties. Start by using the property of logarithms:
\[ \ln\left(\frac{x^6}{8}\right) = \ln(x^6) - \ln(8) \]
Now, differentiate each term:
1. Differentiate \(\ln(x^6)\):
- Use the chain rule: \((\ln(u))' = \frac{1}{u} \cdot u'\),
- Here \(u = x^6\) so \(u' = 6x^5\),
- Thus, \(\frac{d}{dx}[\ln(x^6)] = \frac{1}{x^6} \cdot 6x^5 = \frac{6}{x}\).
2. \(\ln(8)\) is a constant, so its derivative is 0.
The derivative of the function is then:
\[ f'(x) = \frac{6}{x} \]
---
**Result**
\[ f'(x) = \boxed{\frac{6}{x}} \]
Use this result to understand how logarithmic differentiation works and the application of different rules like the chain rule in calculus.
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