system
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human
What form of numeric differentiation is this?

I needed to code up a quick check of a routine that returned the derivative of some function so I found myself doing (f(x+e)-f(x))/e as e got tiny.  So far so good. Then a coworker said that in his experience (f(x+e)-f(x-e))/2e was more accurate for e > 0 because it was symmetric.  I checked that in the limit e-->0 they returned the same derivative for simple functions.  What form of numeric differentiation is this? Is it more accurate?  Thanks! 

EDIT: Lots of great answers in here. Particularly the bit about the central finite difference having no f'' term so the next leading order term scales by e^2 . Thanks y'all!
gpt
There are three simple ways to approximate a derivative, with different amounts of error from the "true" derivative.

d(x) = (f(x+e) - f(x))/e is called a "forward finite difference".  Its error is proportional to e.

d(x) = (f(x) - f(x-e))/e is called a "backward finite difference".  Its error is also proportional to e.

d(x) = (f(x+e)-f(x-e))/(2 e) is called a "central finite difference".  Its error is proportional to e^2 -- which is a big improvement when e is small.

There are other schemes that use f(x+2e), f(x-2e) etc. to create schemes whose error is proportional to e^4 or better, but they're usually not worth the effort, you're better off just making e smaller.