I'm a high school student newly introduced to differentiation, so please excuse my lax use of mathematical notation and/or lack of knowledge of what may well be a basic concept.
We can write a function which produces x! through recursion :
$$f(x)=\begin{cases}x&(x = 1)\\xf(x-1)&(x\gt 1)\end{cases} , x\in \Bbb{N} $$
And "deriving" that using the product and chain rule we get :
$$f'(x)=\begin{cases}1&(x = 1)\\xf'(x-1) + f(x-1)&(x\gt 1)\end{cases}, x\in \Bbb{N}$$
- I am aware that discrete functions aren't differentiable in the proper sense.
I tried to find out some values for f'(x) with a small script, with the following results for the first few values:
| x | f(x) = x! | f'(x) |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 2 | 3 |
| 3 | 6 | 11 |
| 4 | 24 | 50 |
| 5 | 120 | 274 |
| ... | ... | ... |
My question is:
- Does f'(x) correspond to any meaningful mathematical idea?
- If so, is that idea related to a "rate of change" or similar of the factorial function?
Making the problem continuous using the gamma function $$\frac d{dx}\Gamma(x+1)=\Gamma (x+1) \,\psi(x+1)$$ where appears the digamma function. But $$\Gamma (x+1) \,(\psi (x+1)+\gamma )= \Gamma (x+1)\,H_x$$ produces your sequence $$\{1,3,11,50,274,1764,13068,109584,1026576,10628640,\cdots\}$$ which are the unsigned Stirling numbers of first kind that you discovered with a small script.
Congratulations !