What is a discrete-space derivative?

[WhiteRae]

I was looking up how to find the derivative of a factorial and found this thread:
https://www.physicsforums.com/showthread.php?t=1328

What is a discrete-space derivative? I tried looking it up, but had no luck. If someone could explain it in a way someone in calc 2 could understand that would be great, or at least point me in the direction of more information.

 

[VeeEight]

A discrete space is a space where all subsets are open. I have never heard of a discrete space derivative, however.

Have you heard of the gamma function? http://en.wikipedia.org/wiki/Gamma_function
Take a look at how it relates to the factorial function. It does take a lot of work to understand if you're only at the first year level in math.

 

[WhiteRae]

I was looking at the Gamma Function. I understood it a little bit, but in general it was way over my head.

 

[gabbagabbahey]

What is a discrete-space derivative? I tried looking it up, but had no luck. If someone could explain it in a way someone in calc 2 could understand that would be great, or at least point me in the direction of more information.

The general idea is that the more familiar continuous space derivative,

$$\frac{df}{dx}=\lim_{\Delta x\to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$$

Can be generalized in cases where the smallest possible difference between two different values of a discrete variable $x$ is a constant (in other words, all elements of the set are equally spaced) $\delta x$ according to the equation

$$\frac{df}{dx}=\frac{f(x+\delta x)-f(x)}{\delta x}$$

(For a continuous space, $\delta x$ is infinitesimally small)

If $x$ is only allowed to be a natural number, then the smallest diffence between two different values of x is 1, and so

$$\frac{df}{dx}=f(x+1)-f(x)$$

Whether or not this is really the type of derivative you are interested in depends on the context of your problem.