## Problem Statement

Find the derivative.

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

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

## Problem Kuznetsov Differentiation 14-1