Example 1. Find the second-order derivative of the function $y=\sin ^{2} x$.
Solution. Differentiating, we obtain the first derivative

$$
y^{\prime}=2 \sin x \cos x=\sin 2 x
$$

Differentiating again, we find the required second-order derivative:

$$
y^{\prime \prime}=(\sin 2 x)^{\prime}=\cos 2 x \cdot(2 x)^{\prime}=2 \cos 2 x
$$