## Task Condition

Find the derivative.

$y=x^{e^{\sin x}}$
## Solution

$y=x^{e^{\sin x}}$

$\ln y=e^{\sin x} \cdot \ln x$

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

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

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

## Problem Kuznetsov Differentiation 12-25