Answer :

caylus
Hello,

ln(y)=ln(x^(ln(x))=ln(x)*ln(x)=(ln(x))²

(ln(y))'=2ln(x)*1/x
(1/y)*y'=2ln(x) / x
y'=(2 ln(x) * x^(ln(x)) )  /x
Benjamin16
[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]

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