Basics of Derivatives

Differentiation is the tool that turns slopes into numbers. Let's make sure the basic machinery is operational.

Compute the derivative of $f(x)=x^x$ for $x>0$. As a sanity check on your method, also state the value of $f^{\prime }(1)$.

Then briefly explain why a direct application of either the power rule $\left(\frac{d}{dx}{x}^n=n{x}^{n-1}\right)$ or the exponential rule $\left(\frac{d}{dx}{a}^x=a^x\ln a\right)$ fails here, and what technique rescues you.