Finite Difference - Relation With Derivatives

Relation With Derivatives

The derivative of a function f at a point x is defined by the limit

If h has a fixed (non-zero) value instead of approaching zero, then the right-hand side of the above equation would be written

Hence, the forward difference divided by h approximates the derivative when h is small. The error in this approximation can be derived from Taylor's theorem. Assuming that f is continuously differentiable, the error is

The same formula holds for the backward difference:

However, the central difference yields a more accurate approximation. Its error is proportional to square of the spacing (if f is twice continuously differentiable):

The main problem with the central difference method, however, is that oscillating functions can yield zero derivative. If for uneven, and for even, then if it is calculated with the central difference scheme. This is particularly troublesome if the domain of is discrete.