## Task Condition

Find the derivative.

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

$y=(\sin x)^{5 e^{x}}$

$\ln y=5 e^{x} \cdot \ln (\sin x)$

$\frac{y^{\prime}}{y}=\left(5 e^{x} \cdot \ln (\sin x)\right)^{\prime}=5 e^{x} \cdot \ln (\sin x)+5 e^{x} \cdot \frac{1}{\sin x} \cdot \cos x=$

$=5 e^{x} \cdot \ln (\sin x)+5 e^{x} \cdot \operatorname{ctg} x=5 e^{x} \cdot(\ln (\sin x)+\operatorname{ctg} x)$

$y^{\prime}=y \cdot 5 e^{x} \cdot(\ln (\sin x)+\operatorname{ctg} x)=(\sin x)^{5 e^{x}} \cdot 5 e^{x} \cdot(\ln (\sin x)+\operatorname{ctg} x)=$

$=5 e^{x} \cdot(\sin x)^{5 e^{x}} \cdot(\ln (\sin x)+\operatorname{ctg} x)$

## Problem Kuznetsov Differentiation 12-3