## Problem Statement

Find the derivative.

$$
y=x^{3} \arcsin x+\frac{x^{2}+2}{3} \sqrt{1-x^{2}}
$$
## Solution

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

## Problem Kuznetsov Differentiation 14-4