Zero to the Power of Zero/Derivatives

Example of Zero to the Power of Zero
Consider the identity mapping:


 * $I_{\mathbb F} \left({x}\right) = x$

where $\mathbb F \in \left\{ {\R, \C}\right\}$.

From Derivative of Identity Function:


 * $\dfrac {\d I_{\mathbb F} } {\d x} = 1$

But $I_{\mathbb F} \left({x}\right) = x^1$ is also an order one polynomial.

By Power Rule for Derivatives:


 * $\dfrac {\d I_{\mathbb F} } {\d x} = 1 x^0$

As $I_{\mathbb F}$ is differentiable at $0$, for these theorems to be consistent, we insist that $0^0 = 1$.