[tex]y= x^{\ln x} = \left ( e^{\ln x} \right )^{\ln x} = e^{\ln x \cdot \ln x} = e^{\ln^{2} x} \\ \\ y' = \left ( e^{\ln^{2} x} \right )' = e^{\ln^{2} x} \cdot \left ( \ln^{2} x \right )' = e^{\ln^{2} x} \cdot 2 \ln x \cdot (\ln x)' = \\ \\ = \dfrac{ e^{\ln^{2} x} \cdot 2 \ln x }{x} = \boxed{ \dfrac{2\cdot x^{\ln x} \cdot \ln x }{x} }[/tex]
