Example 10. $y=\sin ^{3} x^{2} \cdot \arccos \sqrt{x}$.
Solution. We are dealing with the derivative of a product of composite functions:

$$
\begin{aligned}
& y^{\prime}=\left(\sin ^{3} x^{2}\right)^{\prime} \cdot \arccos \sqrt{x}+\sin ^{3} x^{2}(\arccos \sqrt{x})^{\prime}= \\
& =3 \sin ^{2} x^{2} \cdot \cos x^{2} \cdot 2 x \cdot \arccos \sqrt{x}-\frac{\sin ^{3} x^{2}}{\sqrt{1-x} \cdot 2 \sqrt{x}}= \\
& =6 x \sin ^{2} x^{2} \cdot \cos x^{2} \arccos \sqrt{x}-\frac{\sin ^{3} x^{2}}{2 \sqrt{x(1-x)}}
\end{aligned}
$$

Note. When differentiating functions consisting of a large number of factors or functions of the form $y=f(x)^{g(x)}$, it is recommended to perform preliminary logarithmic differentiation.