## Task Condition

Find the derivative.

$y=x^{e^{x}} \cdot x^{9}$
## Solution

$y=x^{e^{x}} \cdot x^{9}$

$\ln y=\ln \left(x^{e^{x}} \cdot x^{9}\right)=e^{x} \cdot \ln x+9 \cdot \ln x=\ln x \cdot\left(e^{x}+9\right)$

$$
\begin{aligned}
& \frac{y^{\prime}}{y}=\left(\ln x \cdot\left(e^{x}+9\right)\right)^{\prime}=\frac{1}{x} \cdot\left(e^{x}+9\right)+\ln x \cdot e^{x}=e^{x} \cdot\left(\ln x+\frac{1}{x}\right)+\frac{9}{x} \\
& y^{\prime}=y \cdot\left(e^{x} \cdot\left(\ln x+\frac{1}{x}\right)+\frac{9}{x}\right)=x^{e^{x}} \cdot x^{9} \cdot\left(e^{x} \cdot\left(\ln x+\frac{1}{x}\right)+\frac{9}{x}\right)
\end{aligned}
$$

## Problem Kuznetsov Differentiation 12-31