Let f(x) = ln(e* – 5) . Find f'(x). A) f'(x) = 3e3z e3z -5 B) f'(x) = e3z -5 C) f'(x) = -3e3z e3z – 5 -e3z e3z -5 D) f'(x) = E) f'(z) = 굶5 1 e3z -5

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...
icon
Related questions
Question
**Problem Statement:**

Let \( f(x) = \ln(e^{3x} - 5) \). Find \( f'(x) \).

**Options:**

A) \( f'(x) = \frac{3e^{3x}}{e^{3x} - 5} \)

B) \( f'(x) = \frac{e^{3x}}{e^{3x} - 5} \)

C) \( f'(x) = \frac{-3e^{3x}}{e^{3x} - 5} \)

D) \( f'(x) = \frac{-e^{3x}}{e^{3x} - 5} \)

E) \( f'(x) = \frac{1}{e^{3x} - 5} \)

F) Answer not listed

**Correct Answer: A**

**Explanation:**

To find \( f'(x) \), we use the chain rule. The derivative of \( \ln(u) \) is \( \frac{1}{u} \cdot u' \). Here, \( u = e^{3x} - 5 \).

1. Find \( u' = \frac{d}{dx}(e^{3x} - 5) = 3e^{3x} \).
2. Apply the chain rule: \( f'(x) = \frac{1}{e^{3x} - 5} \cdot 3e^{3x} \).
3. Simplify to get \( f'(x) = \frac{3e^{3x}}{e^{3x} - 5} \).

Hence, the correct answer is option A.
Transcribed Image Text:**Problem Statement:** Let \( f(x) = \ln(e^{3x} - 5) \). Find \( f'(x) \). **Options:** A) \( f'(x) = \frac{3e^{3x}}{e^{3x} - 5} \) B) \( f'(x) = \frac{e^{3x}}{e^{3x} - 5} \) C) \( f'(x) = \frac{-3e^{3x}}{e^{3x} - 5} \) D) \( f'(x) = \frac{-e^{3x}}{e^{3x} - 5} \) E) \( f'(x) = \frac{1}{e^{3x} - 5} \) F) Answer not listed **Correct Answer: A** **Explanation:** To find \( f'(x) \), we use the chain rule. The derivative of \( \ln(u) \) is \( \frac{1}{u} \cdot u' \). Here, \( u = e^{3x} - 5 \). 1. Find \( u' = \frac{d}{dx}(e^{3x} - 5) = 3e^{3x} \). 2. Apply the chain rule: \( f'(x) = \frac{1}{e^{3x} - 5} \cdot 3e^{3x} \). 3. Simplify to get \( f'(x) = \frac{3e^{3x}}{e^{3x} - 5} \). Hence, the correct answer is option A.
Expert Solution
Step 1

Calculus homework question answer, step 1, image 1

steps

Step by step

Solved in 2 steps with 2 images

Blurred answer
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
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: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Precalculus
Precalculus
Calculus
ISBN:
9780135189405
Author:
Michael Sullivan
Publisher:
PEARSON
Calculus: Early Transcendental Functions
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning