## Task Condition

Find the derivative.

$y=x^{e^{\operatorname{arctan} x}}$
## Solution

$\ln y=e^{\operatorname{arctg} x} \cdot \ln x$

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

$(\ln y)^{\prime}=\frac{y^{\prime}}{y} \Longrightarrow y^{\prime}=y \cdot(\ln y)^{\prime}$

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

## Problem Kuznetsov Differentiation 12-27