## Task Condition

Find the derivative.

$$
y=(\tan x)^{4 e^{r}}
$$
## Solution

$$
\begin{aligned}
& y=(\tan x)^{4 e^{-x}} \\
& \ln y=4 e^{x} \cdot \ln (\tan x) \\
& \frac{y^{\prime}}{y}=\left(4 e^{x} \cdot \ln (\tan x)\right)^{\prime}=\left(4 e^{-x}\right)^{\prime} \cdot \ln (\tan x)+4 e^{-x} \cdot(\ln (\tan x))^{\prime}= \\
& =4 e^{x} \cdot \ln (\tan x)+4 e^{x} \cdot \frac{1}{\tan x} \cdot(\tan x)^{\prime}=4 e^{x} \cdot \ln (\tan x)+4 e^{x} \cdot \frac{1}{\tan x} \cdot \frac{1}{\cos ^{2} x}= \\
& =4 e^{x} \cdot \ln (\tan x)+4 e^{x} \cdot \frac{1}{\sin x \cdot \cos x}=4 e^{x} \cdot\left(\ln (\tan x)+\frac{2}{\sin 2 x}\right) \\
& y^{\prime}=y \cdot 4 e^{x} \cdot\left(\ln (\tan x)+\frac{2}{\sin 2 x}\right)=(\tan x)^{4 e^{-x}} \cdot 4 e^{x} \cdot\left(\ln (\tan x)+\frac{2}{\sin 2 x}\right)
\end{aligned}
$$