Answer
$$x^{\sqrt{x}}\left(x^{\ln x}\right) \left( \frac{\ln \left(x\right)}{2\sqrt{x}}+\frac{1}{\sqrt{x}}+\frac{2\ln \left(x\right)}{x} \right)$$
Work Step by Step
Given $$y= x^{\sqrt{x}}\left(x^{\ln x}\right)$$
Since
\begin{align*}
\ln y&=\ln x^{\sqrt{x}}\left(x^{\ln x}\right)\\
&=\ln( x^{\sqrt{x}} )+\ln( x^{\ln x} )\\
&= \sqrt{x} \ln( x )+ (\ln x )^2
\end{align*}
Differentiate both sides
\begin{align*}
\frac{1}{y}y'&= \frac{\ln \left(x\right)}{2\sqrt{x}}+\frac{1}{\sqrt{x}}+\frac{2\ln \left(x\right)}{x}\\
y'&=x^{\sqrt{x}}\left(x^{\ln x}\right) \left( \frac{\ln \left(x\right)}{2\sqrt{x}}+\frac{1}{\sqrt{x}}+\frac{2\ln \left(x\right)}{x} \right)
\end{align*}
