## Task Condition

Find the derivative.

$y=(\sin \sqrt{x})^{e^{1 / x}}$
## Solution

$y=(\sin \sqrt{x})^{e^{1 / x}}$

$\ln y=e^{\frac{1}{x}} \cdot \ln (\sin \sqrt{x})$

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

$=e^{\frac{1}{x}} \cdot\left(\frac{\ln (\sin \sqrt{x})}{x}+\frac{\tan \sqrt{x}}{2 \sqrt{x}}\right)$

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

## Problem Kuznetsov Differentiation 12-21