Three point numerical differentiation

Question

Is there any generalized way to calculate numerical differentiation using a certain number of points? I have found 2-point and 5-point methods, but could not find information about using any other number of points. I am interested in doing 3-point, but am not sure if this would be practical or possible.

Answer 1

The general method is as follows.

1. Decide which points you want to use: maybe $x-2h$, $x+h$ and $x+3h$ for some reason. Here $x$ refers to the point at which I want to compute the derivative.
2. Write down Taylor expansions for those points, centered at $x$. Use as many terms as you have points: $$f(x-2h) = f(x) - 2h f'(x) + 2h^2 f''(x)+O(h^3) $$ $$f(x+h) = f(x) + h f'(x) + 0.5 h^2 f''(x)+O(h^3) $$ $$f(x+3h) = f(x) + 3h f'(x) + 4.5 h^2 f''(x)+O(h^3) $$
3. Find a linear combination of these lines that eliminates all derivatives except the one you want, and makes the coefficient of that derivative $1$. This means solving a linear system of $3$ equations with $3$ unknowns, in the above case. If $f'(x)$ is desired, the combination is $$ -\frac{4}{15h}f(x-2h) + \frac1{6h} f(x+h) + \frac{1}{10h} f(x+3h) = f'(x) +O(h^2) $$