If a function f(x) is tabulated at a uniform interval h, then f(x0 + nh) = f(xn) = fn where n is an integer and x0 is any arbitrary value.

First Derivatives

Approximations to f′0, plus the order of the leading error term, are given below.

Centre Differences

Centre difference approximations to the first derivative

Forward Differences

Forward difference approximations to the first derivative

Backward Differences

Backward difference approximations to the first derivative

Second Derivatives

Approximations to f″0, plus the order of the leading error term, are given below.

Centre Differences

Centre difference approximations to the second derivative

Forward Differences

Forward difference approximations to the second derivative

Backward Differences

Backward difference approximations to the second derivative