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 given function is: \[ f(x) = \sin(x^2). \] To find the derivative of \( f(x) \) with respect to \( x \), we use the chain rule: \[ f'(x) = \frac{d}{dx}[\sin(x^2)] = \cos(x^2) \cdot \frac{d}{dx}[x^2]. \] The derivative of \( x^2 \) with respect to \( x \) is: \[ \frac{d}{dx}[x^2] = 2x. \] Thus: \[ f'(x) = \cos(x^2) \cdot 2x. \] At \( x = \sqrt{\pi} \), substitute \( x = \sqrt{\pi} \) into the expression for \( f'(x) \): \[ f'(\sqrt{\pi}) = \cos((\sqrt{\pi})^2) \cdot 2\sqrt{\pi}. \] Simplify \( (\sqrt{\pi})^2 \) to \( \pi \): \[ f'(\sqrt{\pi}) = \cos(\pi) \cdot 2\sqrt{\pi}. \] The value of \( \cos(\pi) \) is \( -1 \). Therefore: \[ f'(\sqrt{\pi}) = -1 \cdot 2\sqrt{\pi} = -2\sqrt{\pi}. \] Hence, the correct answer is (C) \(-2\sqrt{\pi}\).