Differentiate x^x and analyze domain

For $x>0$, differentiate $f(x)=x^x$ by applying logarithmic differentiation. Specify exactly the domain on which the derivative takes real values. Compute $\lim_{x\to 0^+} x^x$ and $f'(1)$. Give a short explanation of whether $f(x)=x^x$ has any critical points on $(0,\infty)$.

Overview: This problem tests skill with calculus ideas including logarithmic differentiation, differentiating functions with variable exponents, evaluating limits at the edge of the domain, and recognizing critical points.