Alternate formula for $Γ(t + 1)$ I believe that:
$$\frac{d^{n}}{dx^{n}}[x^{n}] \equiv Γ(n + 1) \equiv n!$$
Would this have any application, if it has not already been discovered, which I am almost certain that it has?

This is not true, but it is close. The $n$th derivative of $x^n$ is $n!$. This can be proven by induction. 
However $t=n-1$, thus $\Gamma(t+1)=\Gamma(n)=(n-1)!$.  So you want $t=n$ instead.
So it was already known. 
