In mathematics, to approximate a derivative to an arbitrary order of accuracy, it is possible to use the finite difference. A finite difference can be central, forward or backward.
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Central finite difference
This table contains the coefficients of the central differences, for several orders of accuracy and with uniform grid spacing:
For example, the third derivative with a second-order accuracy is
where
Forward and backward finite difference
This table contains the coefficients of the forward differences, for several orders of accuracy and with uniform grid spacing:
For example, the first derivative with a third-order accuracy and the second derivative with a second-order accuracy are
while the corresponding backward approximations are given by
In general, to get the coefficients of the backward approximations, give all odd derivatives listed in the table the opposite sign, whereas for even derivatives the signs stay the same. The following table illustrates this: