## Task Condition

Find the derivative.

$$
y=(\cos 5 x)^{e^{x}}
$$
## Solution

$$
\begin{aligned}
& y=(\cos 5 x')^{e^{e^{x}}} \\
& \ln y=e^{x} \cdot \ln (\cos 5 x) \\
& \frac{y'}{y}=e^{x} \cdot \ln (\cos 5 x)+e^{x} \cdot \frac{1}{\cos 5 x} \cdot(-\sin 5 x)= \\
& =e^{x} \cdot \ln (\cos 5 x)-e^{x} \cdot \tan 5 x=e^{x} \cdot(\ln (\cos 5 x)-\tan 5 x) \\
& y'=y \cdot e^{x} \cdot(\ln (\cos 5 x)-\tan 5 x)=(\cos 5 x)^{e^{x}} \cdot e^{x} \cdot(\ln (\cos 5 x)-\tan 5 x)
\end{aligned}
$$

## Problem Kuznetsov Differentiation 12-10