Question

Find \( \frac{dy}{dx} \) if \[ y = \sqrt{\sin x^2}. \]

Hint:

Use the chain rule carefully when differentiating compositions like \( \sqrt{\sin x^2} \). Differentiate outer function first, then inner function.

Answer 1

Rewrite: \[ y = (\sin x^2)^{1/2}. \] Using the chain rule: \[ \frac{dy}{dx} = \frac{1}{2} (\sin x^2)^{-1/2} \cdot \frac{d}{dx} (\sin x^2). \] Differentiate inside: \[ \frac{d}{dx} (\sin x^2) = \cos x^2 \cdot \frac{d}{dx} (x^2) = \cos x^2 \cdot 2x. \] Therefore, \[ \frac{dy}{dx} = \frac{1}{2} (\sin x^2)^{-1/2} \cdot 2x \cos x^2 = \frac{x \cos x^2}{\sqrt{\sin x^2}}. \]