I’ll go ahead and work with the common 3 point derivative formulas:
This is the three-point endpoint formula: (assuming that has 3 continuous derivatives on the appropriate interval)
where is some point in the interval.
The three point midpoint formula is:
The derivation of these formulas: can be obtained from either using the Taylor series centered at or using the Lagrange polynomial through the given points and differentiating.
That isn’t the point of this note though.
The point: how can one demonstrate, by an example, the role the error term plays.
I suggest trying the following: let vary from, say, 0 to 3 and let . Now use the three point derivative estimates on the following functions:
Note one: the three point estimates for the derivatives will be exactly the same for both and . It is easy to see why.
Note two: the “errors” will be very, very different. It is easy to see why: look at the third derivative term: for it is
The graphs shows the story.
Clearly, the 3 point derivative estimates cannot distinguish these two functions for these “sample values” of , but one can see how in the case of , the degree that wanders away from is directly related to the higher order derivative of .