[tex]f(x)=x^{ex}=e^{ex\ln x}\Rightarrow f'(x)=e^{ex\ln x}\cdot \left ( ex\ln x \right )'=\\={e}^{e\,x\,\ln\left(x\right)+1}\cdot \left(\left(x\right)'\cdot \ln\left(x\right)+\left(\ln\left(x\right)\right)'\cdot x\right)={e}^{e\,x\,\ln\left(x\right)+1}\,\left(\ln\left(x\right)+1\right)[/tex]