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In the limit as goes to zero. This is the formula for the second difference, but now divided by as required from the typical way the second difference vanishes. The difference vanishes as the first power of , and if you were to divide out by , you would get something constant, and the difference of this thing vanishes as . So the second difference goes to zero as the second power of .
You can define third derivatives, and so on. A physicist generally has to be familiar with these discrete forms of the second derivative, since there are many cases, like atomic lattices in a solid, where there is a real, actual , and you are only dealing with an approximate continuum. It is likely that every notion of spatial continuum is related to a limiting quantity which in our universe is large, but finite.
The properties of calculus of finite differences translate to derivatives very simply:
Anyway, going to infinitesimal calculus gives you a few new things: 1. The formulas simplify somewhat, since you are only interested in asymptotics. 2. The derivative gives a meaning to the notion of "how far do you go in an infinitesimal amount of time", and this defines the notion of velocity at a given time. 3. The derivative obeys the chain rule.
The chain rule is a rule for composite functions, f(g(x)). In the discrete case, you couldn't do anything regarding this, because there is no relation between f(g(n+1)) and f(g(n)) that is simple regarding f and g, since the steps g takes might be large. You can write this as but now you are stuck, since is not necessarily an integer.
