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
![### Exercise 14
**Problem:**
Find \( f'(x) \) for the function \( f(x) = \frac{x+9}{x^2 - 7x + 1} \).
**Instructions:**
Apply the quotient rule to find the derivative.
---
**Explanation of Concepts:**
To solve this problem, you need to understand and apply the quotient rule, which is used to differentiate functions that are expressed as the quotient of two functions.
The quotient rule states:
If you have a function \( g(x) = \frac{u(x)}{v(x)} \), then the derivative \( g'(x) \) is given by:
\[
g'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}
\]
Where:
- \( u(x) \) is the numerator function \( x+9 \).
- \( v(x) \) is the denominator function \( x^2 - 7x + 1 \).
- \( u'(x) \) and \( v'(x) \) are the derivatives of \( u(x) \) and \( v(x) \) respectively.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4979c4fb-69aa-4561-8ea9-69969c4f32d0%2Fe9fe518b-e992-497e-a374-94ea321cc34f%2Foe9ewg7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Exercise 14
**Problem:**
Find \( f'(x) \) for the function \( f(x) = \frac{x+9}{x^2 - 7x + 1} \).
**Instructions:**
Apply the quotient rule to find the derivative.
---
**Explanation of Concepts:**
To solve this problem, you need to understand and apply the quotient rule, which is used to differentiate functions that are expressed as the quotient of two functions.
The quotient rule states:
If you have a function \( g(x) = \frac{u(x)}{v(x)} \), then the derivative \( g'(x) \) is given by:
\[
g'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}
\]
Where:
- \( u(x) \) is the numerator function \( x+9 \).
- \( v(x) \) is the denominator function \( x^2 - 7x + 1 \).
- \( u'(x) \) and \( v'(x) \) are the derivatives of \( u(x) \) and \( v(x) \) respectively.
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Quotient Rule of differentiation :
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