1. **Problem statement:** Differentiate the function $y = \sin^2 x$ with respect to $x$. 2. **Formula and rules:** Use the chain rule for differentiation. If $y = [f(x)]^2$, then $\frac{dy}{dx} = 2 f(x) \cdot f'(x)$. 3. **Apply the chain rule:** Here, $f(x) = \sin x$, so $f'(x) = \cos x$. 4. **Calculate the derivative:** $$\frac{dy}{dx} = 2 \sin x \cdot \cos x$$ 5. **Simplify the expression:** Using the double-angle identity $\sin 2x = 2 \sin x \cos x$, we get $$\frac{dy}{dx} = \sin 2x$$ **Final answer:** $$\frac{d}{dx} \sin^2 x = \sin 2x$$