## Problem Statement

Find the derivative.

$y=\arcsin \frac{\sqrt{x}-2}{\sqrt{5 x}}$
## Solution

$$
\begin{aligned}
& y^{\prime}=\left(\arcsin \frac{\sqrt{x}-2}{\sqrt{5 x}}\right)^{\prime}=\frac{1}{\sqrt{1-\left(\frac{\sqrt{x}-2}{\sqrt{5 x}}\right)^{2}}} \cdot\left(\frac{\sqrt{x}-2}{\sqrt{5 x}}\right)^{\prime}= \\
& =\frac{\sqrt{5 x}}{\sqrt{5 x-(\sqrt{x}-2)^{2}}} \cdot \frac{(\sqrt{x}-2)^{\prime} \cdot \sqrt{5 x}-(\sqrt{x}-2) \cdot(\sqrt{5 x})^{\prime}}{5 x}= \\
& =\frac{\sqrt{5 x}}{\sqrt{5 x-x+4 \sqrt{x}-4}} \cdot \frac{\frac{1}{2 \sqrt{x}} \cdot \sqrt{5 x}-(\sqrt{x}-2) \cdot \frac{1}{2 \sqrt{5 x}} \cdot 5}{5 x}= \\
& =\frac{5 x-(\sqrt{x}-2) \cdot \sqrt{x} \cdot 5}{10 x \sqrt{x} \sqrt{4 x+4 \sqrt{x}-4}}=\frac{5 x-5 x+10 \sqrt{x}}{10 x \sqrt{x} \sqrt{4 x+4 \sqrt{x}-4}}= \\
& =\frac{10 \sqrt{x}}{10 x \sqrt{x} \sqrt{4 x+4 \sqrt{x}-4}}=\frac{1}{2 x \sqrt{x+\sqrt{x}-1}}
\end{aligned}
$$

## Problem Kuznetsov Differentiation 10-2