Zero to the Power of Zero/Derivatives

Example of Zero to the Power of Zero

Consider the identity mapping:

$\map {I_\GF} x = x$

where $\GF \in \set {\R, \C}$.

From Derivative of Identity Function:

$\dfrac {\d I_\GF} {\d x} = 1$

But $\map {I_\GF} x = x^1$ is also an order one polynomial.

By Power Rule for Derivatives:

$\dfrac {\d I_\GF} {\d x} = 1 x^0$

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