I am stuck at this! Differentiate with respect to x:
y = ( ln x ) to the power ln x
Thanks
$\displaystyle y=(\ln(x))^{\ln(x)}$
This is going to need the chain rule and this identity
$\displaystyle \ln(x)^{\ln(x)}=e^{ln(\ln(x)^{\ln(x)})}=e^{\ln(x)\ cdot \ln(\ln(x))}$
So now we get
$\displaystyle y'=e^{\ln(x)\cdot \ln(\ln(x))}\left( \frac{1}{x}\ln(\ln(x)+\ln(x)\left(\frac{1}{\ln(x)} \frac{1}{x}\right)\right)$
$\displaystyle y'=\ln(x)^{\ln(x)}\left( \frac{\ln(\ln(x))}{x}+\frac{1}{x}\right)$