Question

Let \( f(x) = x \sin(x^4) \). Then \( f'(x) \) at \( x = \sqrt[4]{\pi} \) is equal to:

• A. \( 4\pi + 1 \)
• B. \( 4\pi \)
• C. \( -4\pi \)
• D. \( 4\pi - 1 \)
• E. \( 4\pi + 4 \)

Hint:

When applying the product rule, remember to distribute the derivative to each part of the product and simplify the expression before substituting values.

Answer 1

The Correct Option is C

First, find the derivative \( f'(x) \) using the product rule: \[ f(x) = x \sin(x^4) \] \[ f'(x) = \sin(x^4) \cdot \frac{d}{dx}[x] + x \cdot \frac{d}{dx}[\sin(x^4)] \] \[ f'(x) = \sin(x^4) + x \cos(x^4) \cdot 4x^3 \] \[ f'(x) = \sin(x^4) + 4x^4 \cos(x^4) \] Now, substitute \( x = \sqrt[4]{\pi} \) into \( f'(x) \): \[ f'(\sqrt[4]{\pi}) = \sin((\sqrt[4]{\pi})^4) + 4(\sqrt[4]{\pi})^4 \cos((\sqrt[4]{\pi})^4) \] \[ = \sin(\pi) + 4\pi \cos(\pi) \] \[ = 0 + 4\pi \cdot (-1) \] \[ = -4\pi \] Thus, \( f'(x) \) evaluated at \( x = \sqrt[4]{\pi} \) is \( -4\pi \).