Verified 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

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

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