any pattern here ? (revised 2)
I see finite difference representations of derivatives. http://mathworld.wolfram.com/FiniteDifference.html
As an engineering-type, I think Wolfram does a crappy job for it.
I prefer a numerical methods text with good tables. http://amzn.com/0072431938 Try up in the 630's. 626 perhaps? I'm doing this from memory.
Proper derivation is done using sums of Taylor series approximations to the function.
Aha. Less bad. https://reference.wolfram.com/mathematica/tutorial/NDSolvePDE.html Check out the large table about 10% down the page.