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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![**Problem Statement:**
Given the function \( f(x) = \ln(x^2 - 14x + 52) \).
**Objective:**
Find the derivative of the function, denoted as \( f'(x) \).
**Solution Approach:**
To find the derivative \( f'(x) \), you will apply the chain rule. The chain rule states that the derivative of \( \ln(u) \) is \( \frac{1}{u} \cdot \frac{du}{dx} \).
1. Identify the inner function \( u(x) = x^2 - 14x + 52 \).
2. Differentiate \( u(x) \) with respect to \( x \):
- \( \frac{du}{dx} = 2x - 14 \).
3. Apply the chain rule:
- \( f'(x) = \frac{1}{x^2 - 14x + 52} \cdot (2x - 14) \).
Thus, the derivative of the function is:
\[ f'(x) = \frac{2x - 14}{x^2 - 14x + 52} \]
**Make sure to review and simplify your expression before finalizing the derivative.**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe345ae0b-fd83-4d2e-a467-23dbb110ff23%2F7224c45f-d8ef-4df4-b149-0f4888d17c06%2Frygilm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Given the function \( f(x) = \ln(x^2 - 14x + 52) \).
**Objective:**
Find the derivative of the function, denoted as \( f'(x) \).
**Solution Approach:**
To find the derivative \( f'(x) \), you will apply the chain rule. The chain rule states that the derivative of \( \ln(u) \) is \( \frac{1}{u} \cdot \frac{du}{dx} \).
1. Identify the inner function \( u(x) = x^2 - 14x + 52 \).
2. Differentiate \( u(x) \) with respect to \( x \):
- \( \frac{du}{dx} = 2x - 14 \).
3. Apply the chain rule:
- \( f'(x) = \frac{1}{x^2 - 14x + 52} \cdot (2x - 14) \).
Thus, the derivative of the function is:
\[ f'(x) = \frac{2x - 14}{x^2 - 14x + 52} \]
**Make sure to review and simplify your expression before finalizing the derivative.**
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