I have a theory that uses the gamma function:
$$\Gamma(n)=\int_0^\infty x^{n-1}e^{-x} \space dx$$
Then I was inclined to think that perhaps the derivative is:
$$x^{n-1}e^{-x}$$
But I'm not sure we can just drop the integral along with the bounds to get the derivative. Then I thought about taking the limit:
$$\lim_{x\to\infty}x^{n-1}e^{-x}$$
But now we can't specify at what $x$ value we want to get the rate of change of. At this point I feel like I can't get any further on my own and would appreciate some insight.
EDIT: Looking for derivative in terms of $n$ actually.