## Task Condition

Find the derivative.

$y=(\sin \sqrt{x})^{\ln (\sin \sqrt{x})}$
## Solution

$y=(\sin \sqrt{x})^{\ln (\sin \sqrt{x})}$

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

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

$=\frac{\ln (\sin \sqrt{x}) \cdot \operatorname{ctg} \sqrt{x}}{\sqrt{x}}$

$y^{\prime}=y \cdot \frac{\ln (\sin \sqrt{x}) \cdot \operatorname{ctg} \sqrt{x}}{\sqrt{x}}=(\sin \sqrt{x})^{\ln (\sin \sqrt{x})} \cdot \frac{\ln (\sin \sqrt{x}) \cdot \operatorname{ctg} \sqrt{x}}{\sqrt{x}}$

## Problem Kuznetsov Differentiation 12-2