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...
Related questions
Question
![**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.
Expert Solution

This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 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