Estimating Second Derivatives Numerically

Suppose that we want to estimate the second derivative of a function f at a point x and that we know the values of the function at three points

(x - h, f(x - h)), (x, f(x)), and (x + h, f(x + h)).

Since f''(x) is the derivative of the function f' at the point x we can estimate f''(x) by

We can estimate f'(x - h/2) and f'(x + h/2) by

and putting this all together we obtain the following estimate for f''(x).

The same idea is used to estimate the second order partial derivatives below.