Question: The derivative of \( f(x) = x^4 \sin(x) \) is \( \frac{d}{dx} f(x) = -x^4 \cos(x) + 4x^3…

The derivative of \( f(x) = x^4 \sin(x) \) is \( \frac{d}{dx} f(x) = -x^4 \cos(x) + 4x^3 \sin(x) \).

Solution

To find the derivative of \( f(x) = x^4 \sin(x) \), we’ll use the product rule. The product rule states: \[ (fg)’ = f’g + fg’ \] Here, let \( f(x) = x^4 \) and \( g(x) = \sin(x) \). First, find the derivative of \( f(x) = x^4 \): \[ f’(x) = 4x^3 \] Next, find the derivative of \( g(x) = \sin(x) \): \[ g’(x) = \cos(x) \] Apply the product rule: \[ \frac{d}{dx}[x^4 \sin(x)] = (4x^3)(\sin(x)) + (x^4)(\cos(x)) \] Thus, the derivative is: \[ = 4x^3 \sin(x) + x^4 \cos(x) \] The provided answer is: \[ - x^4 \cos(x) + 4x^3 \sin(x) \] This indicates that the terms have the opposite signs. The correct derivative based on the product rule calculations is: \[ 4x^3 \sin(x) + x^4 \cos(x) \]