## Task Condition

Find the derivative.

$y=(\sin x)^{\frac{\varsigma_{x}}{2}}$
## Solution

$y=(\sin x)^{\frac{5 x}{2}}$

$\ln y=\ln \left((\sin x)^{\frac{5 x}{2}}\right)=\frac{5 x}{2} \cdot \ln (\sin x)$

$\frac{y^{\prime}}{y}=\left(\frac{5 x}{2} \cdot \ln (\sin x)\right)^{\prime}=\frac{5}{2} \cdot \ln (\sin x)+\frac{5 x}{2} \cdot \frac{1}{\sin x} \cdot \cos x=\frac{5}{2} \cdot(\ln (\sin x)+x \cdot \operatorname{ctg} x)$

$$
\begin{aligned}
& y^{\prime}=y \cdot \frac{5}{2} \cdot(\ln (\sin x)+x \cdot \operatorname{ctg} x)=(\sin x)^{\frac{5 x}{2}} \cdot \frac{5}{2} \cdot(\ln (\sin x)+x \cdot \operatorname{ctg} x)= \\
& =\frac{5}{2} \cdot(\sin x)^{\frac{5 x}{2}} \cdot(\ln (\sin x)+x \cdot \operatorname{ctg} x)
\end{aligned}
$$

## Problem Kuznetsov Differentiation 12-17