## Problem Statement

Find the derivative.

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

$$
\begin{aligned}
& y^{\prime}=\left(\frac{2 \sqrt{1-x} \cdot \arcsin \sqrt{x}}{x}+\frac{2}{\sqrt{x}}\right)^{\prime}=\left(2 \sqrt{\frac{1-x}{x^{2}}} \cdot \arcsin \sqrt{x}+\frac{2}{\sqrt{x}}\right)^{\prime}= \\
& =2 \cdot \frac{1}{2 \sqrt{\frac{1-x}{x^{2}}}} \cdot\left(\frac{-1 \cdot x^{2}-(1-x) \cdot 2 x}{x^{4}}\right) \cdot \arcsin \sqrt{x}+2 \sqrt{\frac{1-x}{x^{2}}} \cdot \frac{1}{\sqrt{1-(\sqrt{x})^{2}}} \cdot \frac{1}{2 \sqrt{x}}+\left(-\frac{1}{2}\right) \cdot \frac{2}{\sqrt{x^{3}}}= \\
& =\frac{x}{\sqrt{1-x}} \cdot \frac{-x-2+2 x}{x^{3}} \cdot \arcsin \sqrt{x}+\frac{\sqrt{1-x}}{x} \cdot \frac{1}{\sqrt{1-x}} \cdot \frac{1}{\sqrt{x}}-\frac{1}{\sqrt{x^{3}}}= \\
& =\frac{x-2}{x^{2} \sqrt{1-x}} \cdot \arcsin \sqrt{x}+\frac{1}{x \sqrt{x}}-\frac{1}{x \sqrt{x}}=\frac{x-2}{x^{2} \sqrt{1-x}} \cdot \arcsin \sqrt{x}
\end{aligned}
$$

Kuznetsov Differentiation 10-19