Not defined in logarithmic differentiation The derivative of $\ln x^x$ by logarithmic differentiation is $\ln x^x[\ln(\ln(x)+(1/\ln x)]$ but what will be derivative when $x=1$ and at small values like $0.00001$ because $\ln (x)$ is negative and $\ln (\ln (x))$ is not defined.

When you are taking logarithm both sides, it is assumed both sides are positive, because of the domain of $ln()$.
Now find out when $\ln\left(x^{x}\right)$ is positive. or $x^{x}>1$
