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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![# Calculus Multiple Choice Question
## Problem Statement
Evaluate the derivative of \( x^3 \sin x \).
\[ \frac{d}{dx} \left[ x^3 \sin x \right] \]
### Options:
- (A) \( 3x^2 \cos x \)
- (B) \( 3x^2 \sin x \)
- (C) \( 3x^2 \sin x + x^3 \cos x \)
- (D) \( x^3 \cos x \)
- (E) None of the above.
### Explanation:
To solve this, use the product rule of differentiation which states:
\[ \frac{d}{dx} [u \cdot v] = \frac{du}{dx} \cdot v + u \cdot \frac{dv}{dx} \]
Here, \( u = x^3 \) and \( v = \sin x \).
First, compute the derivatives of \( u \) and \( v \):
\[ \frac{du}{dx} = 3x^2 \]
\[ \frac{dv}{dx} = \cos x \]
Apply the product rule:
\[ \frac{d}{dx} [x^3 \sin x] = (3x^2)(\sin x) + (x^3)(\cos x) \]
Hence, the correct answer is:
\[ \boxed{\text{(C) } 3x^2 \sin x + x^3 \cos x} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8dc655a5-0114-4983-becd-0bcc5b4edffa%2F1e18b142-1d41-4d24-af2b-b0ccdcad92a4%2Fimwuk04_processed.png&w=3840&q=75)
Transcribed Image Text:# Calculus Multiple Choice Question
## Problem Statement
Evaluate the derivative of \( x^3 \sin x \).
\[ \frac{d}{dx} \left[ x^3 \sin x \right] \]
### Options:
- (A) \( 3x^2 \cos x \)
- (B) \( 3x^2 \sin x \)
- (C) \( 3x^2 \sin x + x^3 \cos x \)
- (D) \( x^3 \cos x \)
- (E) None of the above.
### Explanation:
To solve this, use the product rule of differentiation which states:
\[ \frac{d}{dx} [u \cdot v] = \frac{du}{dx} \cdot v + u \cdot \frac{dv}{dx} \]
Here, \( u = x^3 \) and \( v = \sin x \).
First, compute the derivatives of \( u \) and \( v \):
\[ \frac{du}{dx} = 3x^2 \]
\[ \frac{dv}{dx} = \cos x \]
Apply the product rule:
\[ \frac{d}{dx} [x^3 \sin x] = (3x^2)(\sin x) + (x^3)(\cos x) \]
Hence, the correct answer is:
\[ \boxed{\text{(C) } 3x^2 \sin x + x^3 \cos x} \]
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