## Task Condition

Find the derivative.

$y=5 x-\ln \left(1+\sqrt{1-e^{10 x}}\right)-e^{-5 x} \cdot \arcsin \left(e^{5 x}\right)$
## Solution

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

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

## Problem Kuznetsov Differentiation 13-13