What is the derivative of ##lnx^lnx##?

The derivative of a logarithm (be it natural or base-10) ##ln[f(x)]## is given by the formula:

f'(x) / f(x)

Where f'(x) is the derivative of the original function f(x).

Ok, keep that aside for a while.

Now, going back to logarithms properties, we have that ##ln(x)^m## = ##m.ln(x)##

Then, we have in your example:

##ln(x)^ln(x)## = ##ln(x)## . ##ln(x)##

Deriving it consists on deriving a product, which consists on:

Derivative of the first function times the original second function PLUS Derivative of the second function times the original first function

Like this:

##1/x . ln(x)## + ##ln(x) + 1/x## = ##2.ln(x)/x##

:)