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...
Related questions
Question
![### Problem Statement:
Given:
\[ f(x) = 5 \cos(6 \ln(x)) \]
Tasks:
1. Find the derivative \( f'(x) \).
2. Evaluate the derivative at \( x = 2 \), denoted as \( f'(2) \).
### Instructions:
- Use the chain rule for differentiation to solve for \( f'(x) \).
- After finding \( f'(x) \), substitute \( x = 2 \) to find \( f'(2) \).
### Solution Steps:
1. **Find \( f'(x) \):**
- Identify the outer function \( \cos(u) \) where \( u = 6 \ln(x) \).
- Differentiate the outer function: The derivative of \( \cos(u) \) is \(-\sin(u)\).
- Multiply by the derivative of the inner function \( u = 6 \ln(x) \).
- Differentiate \( 6 \ln(x) \): The derivative is \( \frac{6}{x} \).
- Combine the results using the chain rule.
2. **Calculate \( f'(2) \):**
- Substitute \( x = 2 \) into the expression for \( f'(x) \).
### Note:
- Ensure to provide intermediate steps for a clear understanding of the chain rule application.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3ef879d6-d05a-4de3-b7a5-e661a4d5bec1%2Fa082042c-e2d3-4b31-b63d-5257984cac7e%2Fbzss8c9_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement:
Given:
\[ f(x) = 5 \cos(6 \ln(x)) \]
Tasks:
1. Find the derivative \( f'(x) \).
2. Evaluate the derivative at \( x = 2 \), denoted as \( f'(2) \).
### Instructions:
- Use the chain rule for differentiation to solve for \( f'(x) \).
- After finding \( f'(x) \), substitute \( x = 2 \) to find \( f'(2) \).
### Solution Steps:
1. **Find \( f'(x) \):**
- Identify the outer function \( \cos(u) \) where \( u = 6 \ln(x) \).
- Differentiate the outer function: The derivative of \( \cos(u) \) is \(-\sin(u)\).
- Multiply by the derivative of the inner function \( u = 6 \ln(x) \).
- Differentiate \( 6 \ln(x) \): The derivative is \( \frac{6}{x} \).
- Combine the results using the chain rule.
2. **Calculate \( f'(2) \):**
- Substitute \( x = 2 \) into the expression for \( f'(x) \).
### Note:
- Ensure to provide intermediate steps for a clear understanding of the chain rule application.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 1 images
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning