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. \(2\sqrt{\pi}\)
• E.

Hint:

The chain rule is crucial for differentiating composite functions like \( \sin(x^2) \). Always differentiate the outer function first, then multiply by the derivative of the inner function.

Answer 1

The Correct Option is C

The function provided is: \( f(x) = \sin(x^2) \). To compute its derivative \( f'(x) \), the chain rule is applied: \( f'(x) = \frac{d}{dx}[\sin(x^2)] = \cos(x^2) \cdot \frac{d}{dx}[x^2] \). The derivative of \( x^2 \) with respect to \( x \) is \( \frac{d}{dx}[x^2] = 2x \). Consequently, \( f'(x) = \cos(x^2) \cdot 2x \). Evaluating at \( x = \sqrt{\pi} \): \( f'(\sqrt{\pi}) = \cos((\sqrt{\pi})^2) \cdot 2\sqrt{\pi} \). Simplifying \( (\sqrt{\pi})^2 \) to \( \pi \), we get \( f'(\sqrt{\pi}) = \cos(\pi) \cdot 2\sqrt{\pi} \). Since \( \cos(\pi) = -1 \), the final result is \( f'(\sqrt{\pi}) = -1 \cdot 2\sqrt{\pi} = -2\sqrt{\pi} \). Therefore, the correct answer is (C) \(-2\sqrt{\pi}\).