I've read in several places that defining $0^0=1$ is convenient in several (primarily discrete) settings. One argument on Wikipedia in favor of this definition was the need of a special case for the product rule $\frac{d}{dx} x^n = nx^{n-1}$ for $n=1$ at $x=0$.
Regardless of which definition, if any, is proper, is this not an argument to the contrary? Or I'm missing something? $nx^{n-1}$ for $n=1$ at $x=0$ is $1*0^0 = 1*1 = 1$ contradicts $\frac{d}{dx} 0^1 = \frac{d}{dx} 0 = 0$.
I feel as though there's something just incredibly obvious I'm missing here. How can this be an argument in favor of defining $0^0=1$?