Question

What is the derivative of y = x^(lnx)? Show your solution.

Answer 1

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

Answer 2

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