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](https://lynneslair.com/wp-content/uploads/2021/06/1derivCentre.png)
Forward Differences
![Forward difference approximations to the first derivative](https://lynneslair.com/wp-content/uploads/2021/06/1derivForward.png)
Backward Differences
![Backward difference approximations to the first derivative](https://lynneslair.com/wp-content/uploads/2021/06/1derivBackward.png)
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](https://lynneslair.com/wp-content/uploads/2021/06/2derivCentre.png)
Forward Differences
![Forward difference approximations to the second derivative](https://lynneslair.com/wp-content/uploads/2021/06/2derivForward.png)
Backward Differences
![Backward difference approximations to the second derivative](https://lynneslair.com/wp-content/uploads/2021/06/2derivBackward.png)