Calculus over integers for derivative/integral of factorial?

One usually introduces the Gamma function to define a derivative of the factorial. However couldn't one define a derivative over integers like $$f'(n) = \frac{f(n+1) - f(n)}{1} ?$$ Such a discrete alternative to the usual derivative would then allow us to simply compute a derivative of the factorial like $$(n!)' = (n+1)! - n! = (n+1)n!-n! = n\ n!$$ My question is now: Is this the correct way to define a derivative over integer numbers and is the result for the derivative of the factorial correct? And how would one then compute the discrete integral/sum for the factorial?

• Knuth's Concrete Mathematics has a lot about this difference operator. – Matthew Leingang Feb 17 '15 at 21:29
• If you are interested in other ways to calculating the derivative of the factorial, see here: math.stackexchange.com/questions/1633014/… Another note is that taking the derivative of Stirling's approximations might be a good approach, if we are to avoid the Gamma function. – Simply Beautiful Art Jun 30 '16 at 16:07