Consider the familiar Taylor series expansion:

Now, suppose you know the value of the function f(x) at a variety of different discrete locations, as indicated by the red dots in the figure, above, and you want to know ...

- the first derivative of the function at location x
_{0}, i.e., [ ¶_{x}f ]_{0}.- Derive an approximate expression using a 2-point FD scheme.
- Derive an approximate expression using a 3-point FD scheme.

- the second derivative of the
function at location x
_{0}, i.e., [ ¶_{x}^{2}f ]_{0}.