Find the derivative of the function f:(0,oo)->R, f(x)=x^x.

The domain of the function allows us to write f(x)=x^x  as f(x)=e^{ln(x^x)}=e^{x ln(x)} (since ln(x) is defined on (0,oo) only). Using the standard derivative rules we get f'(x)=e^{x ln(x)} (x ln(x))'=e^{x ln(x)}(1+ln(x))=x^x (1+ln(x)).