Question

The derivative of \( \sin(x^2) \) w.r.t. \( x \), at \( x = \sqrt{\pi} \), is:

• A. \( 1 \)
• B. \( -1 \)
• C. \( -2\sqrt{\pi} \)
• D. \( 2\sqrt{\pi} \)

Hint:

For composite functions like \( \sin(g(x)) \), use the chain rule: differentiate the outer function and multiply by the derivative of the inner function.

Answer 1

The Correct Option is C

Given the function \( f(x) = \sin(x^2) \). Differentiating \( f(x) \) with respect to \( x \) yields: \[ \frac{d}{dx} [\sin(x^2)] = \cos(x^2) \cdot \frac{d}{dx}(x^2) = \cos(x^2) \cdot 2x. \] Evaluating the derivative at \( x = \sqrt{\pi} \): \[ \frac{d}{dx} [\sin(x^2)] = 2x \cos(x^2). \] Substituting \( x = \sqrt{\pi} \) gives: \[ \frac{d}{dx} [\sin(x^2)] = 2\sqrt{\pi} \cos(\pi). \] As \( \cos(\pi) = -1 \), the derivative becomes: \[ \frac{d}{dx} [\sin(x^2)] = 2\sqrt{\pi} \cdot (-1) = -2\sqrt{\pi}. \] The correct answer is (C) \( -2\sqrt{\pi} \).