5 – x 3. If f(x) = , then f'(x) = x' + 2 – 4x³ + 15x² – 2 (A) (² + 2)° – 2r° + 15x² + 2 (B) 2.x³ – 15x² – 2 (C) (* + 2° 4x – 15x2 + 2 (D) (? +2) 2 Page
5 – x 3. If f(x) = , then f'(x) = x' + 2 – 4x³ + 15x² – 2 (A) (² + 2)° – 2r° + 15x² + 2 (B) 2.x³ – 15x² – 2 (C) (* + 2° 4x – 15x2 + 2 (D) (? +2) 2 Page
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 3: Differentiation of a Rational Function**
Given the function \( f(x) = \frac{5 - x}{x^3 + 2} \), find the derivative \( f'(x) \).
**Options:**
- (A) \( \frac{-4x^3 + 15x^2 - 2}{(x^3 + 2)^2} \)
- (B) \( \frac{-2x^3 + 15x^2 + 2}{(x^3 + 2)^2} \)
- (C) \( \frac{2x^3 - 15x^2 - 2}{(x^3 + 2)^2} \)
- (D) \( \frac{4x^3 - 15x^2 + 2}{(x^3 + 2)^2} \)
**Solution Approach:**
To solve this, apply the quotient rule:
\[ f'(x) = \frac{(g(x) \cdot h'(x) - h(x) \cdot g'(x))}{(h(x))^2} \]
For \( f(x) = \frac{g(x)}{h(x)} \), where:
- \( g(x) = 5 - x \)
- \( h(x) = x^3 + 2 \)
Find \( g'(x) \) and \( h'(x) \):
- \( g'(x) = -1 \)
- \( h'(x) = 3x^2 \)
Substitute these into the quotient rule to find \( f'(x) \).
**Explanation of Graphs/Diagrams:**
In a real educational setting, visual aids such as graphs can provide further insight into the behavior of the function and its derivative over certain intervals of \( x \).
**Further Exploration:**
Review and practice the quotient rule for differentiating rational functions and explore additional problems to strengthen understanding of derivatives.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3a27ec27-3428-4e40-a574-a317cad8784a%2F69adf149-b6cd-464e-bb0a-695cf85da195%2Fpir0ew_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 3: Differentiation of a Rational Function**
Given the function \( f(x) = \frac{5 - x}{x^3 + 2} \), find the derivative \( f'(x) \).
**Options:**
- (A) \( \frac{-4x^3 + 15x^2 - 2}{(x^3 + 2)^2} \)
- (B) \( \frac{-2x^3 + 15x^2 + 2}{(x^3 + 2)^2} \)
- (C) \( \frac{2x^3 - 15x^2 - 2}{(x^3 + 2)^2} \)
- (D) \( \frac{4x^3 - 15x^2 + 2}{(x^3 + 2)^2} \)
**Solution Approach:**
To solve this, apply the quotient rule:
\[ f'(x) = \frac{(g(x) \cdot h'(x) - h(x) \cdot g'(x))}{(h(x))^2} \]
For \( f(x) = \frac{g(x)}{h(x)} \), where:
- \( g(x) = 5 - x \)
- \( h(x) = x^3 + 2 \)
Find \( g'(x) \) and \( h'(x) \):
- \( g'(x) = -1 \)
- \( h'(x) = 3x^2 \)
Substitute these into the quotient rule to find \( f'(x) \).
**Explanation of Graphs/Diagrams:**
In a real educational setting, visual aids such as graphs can provide further insight into the behavior of the function and its derivative over certain intervals of \( x \).
**Further Exploration:**
Review and practice the quotient rule for differentiating rational functions and explore additional problems to strengthen understanding of derivatives.
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