## Task Condition

Find the derivative.

$y=(\cos 2 x)^{\ln (\cos 2 x) / 4}$
## Solution

$y=(\cos 2 x)^{\ln (\cos 2 x) / 4}$

$\ln y=\frac{\ln (\cos 2 x)}{4} \cdot \ln (\cos 2 x)$

$\ln y=\frac{\ln ^{2}(\cos 2 x)}{4}$

$\frac{y^{\prime}}{y}=\left(\frac{\ln ^{2}(\cos 2 x)}{4}\right)^{\prime}=\frac{2 \ln (\cos 2 x)}{4} \cdot \frac{1}{\cos 2 x} \cdot(-\sin 2 x) \cdot 2=-\operatorname{tg} 2 x \cdot \ln (\cos 2 x)$

$y^{\prime}=y \cdot(-\operatorname{tg} 2 x \cdot \ln (\cos 2 x))=-(\cos 2 x)^{\ln (\cos 2 x) / 4} \cdot \operatorname{tg} 2 x \cdot \ln (\cos 2 x)$

## Problem Kuznetsov Differentiation 12-30