Derivative of x!

Date: 05/27/2003 at 15:13:49
From: Steven S.
Subject: Derivative of x!

My Pre-Calculus teacher gave my class several derivative practice 
problems to do in our spare time, one of which is:

   (d/dx)(x!)

My initial response was to think that this was not differentiable, 
since it is not a continuous function (this assumption, I believe, is 
wrong). I then generated a table of the (x,y) values for x!, and it 
seems that the slope always comes out to (x*x!). (I'm not sure if this 
is right). To try to prove myself right, I tried to prove the 
derivative using the derivative rules (no luck), then with the 
algebraic definition of the derivative (I got quite stuck). My current 
plan is to try a set-up using series, but I believe that there must be 
an easier way.

Date: 05/28/2003 at 13:57:19
From: Doctor Schwa
Subject: Re: Derivative of x!

>My initial response was to think that this was not differentiable, 
>since it is not a continuous function (this assumption, I believe, is 
>wrong).

Actually, I think you're right. SOMEHOW you're going to have to decide 
how to fill in between the "dots" of the factorial function.

>I then generated a table of the (x,y) values for x!, and it seems 
>that the slope always comes out to (x*x!). (I'm not sure if this is 
>right).

This is a slope for DIFFERENCES: that is, ((x+h)! - x!) / h
when h = 1, not in the limit as h -> 0, as you would need for a 
derivative.

Finite differences (like what you talk about), for functions defined
on the integers, are quite interesting: but not, I think, quite what 
you were looking for here.

Nonetheless, this is a great discovery you've made, and if you're 
curious to explore it further, searching for the phrase "finite 
difference" should find you some useful stuff.  Or let me know and 
I'll send you some good references.

In order to take a true derivative, you have to make x! continuous
somehow. Maybe this discussion from the Dr. Math archives will help:

   The Gamma Function and Its Derivative
   http://mathforum.org/library/drmath/view/53664.html 

Let me know if there's anything more I can do for you.  I
admire your hard work on this problem, and your creative
ideas as well!

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/