Does the derivative of factorial x! make any sense if x is only positive integers?
The usual calculus derivative is really measuring a continuous sort of thing so it doesn’t really make sense in a discrete variable setting. However, there are two ways to make sense of your question. The first is to extend the factorial function so that it covers all the real numbers. The usual way to do this is way the Γ (uppercase gamma) function. The other is to use the discrete derivative. Feel free to look up both. There are extensive references of each on the web.