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Multiple choice. 3 Part C. d [x³ sin x] dx (A) 3x² cos x (D) x³ cos x 3 2 (B) 3x sin x (E) None of the above. 2 3 (C) 3x² sin x + x³ cos x

Multiple choice. 3 Part C. d [x³ sin x] dx (A) 3x² cos x (D) x³ cos x 3 2 (B) 3x sin x (E) None of the above. 2 3 (C) 3x² sin x + x³ cos x

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter3: The Derivative
Section3.5: Graphical Differentiation
Problem 2E
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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: \[ \boxed{\text{(C) } 3x^2 \sin x + x^3 \cos x} \]
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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