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 x₀, i.e., [ ¶_xf ]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 x₀, i.e., [ ¶_x² f ]₀.