The numerical differentiation formulas shown below were calculated using the algorithm from Calculating derivatives of a function numerically. In these equations, h is the distance between sequential x values.

Derivatives can be numerically calculated from sets of equally spaced function values.

For example, for f=x^{2}, consider the points below:

X Y

0 0

1 1

2 4

3 9

For this series, there are 4 pairs for x and y sets. To numerically calcuate the first derivative at x=2, the equation shown below for the first derivative of the third point in a four point series is used.

f^{1} = (2f_{3} + 3f_{2} - 6f_{1} + f_{0})/(6h) = (2*9 + 3*4 - 6*1 + 1*0)/(6*1)= 24/6 = 4

The analytic derivative of x^{2} is 2x. At x = 2, this also gives 2*2 = 4.

Two points first derivatives.

Three points first derivatives.

Three points second derivatives.

Four points first derivatives.

Four points second derivatives.

Four points third derivatives.

Five points first derivatives.

Five points second derivatives.

Five points third derivatives.

Five points fourth derivatives.

Six points first derivatives.

Six points second derivatives.

Six points third derivatives.

Six points fourth derivatives.

Six points fifth derivatives.

Seven points first derivatives.

Seven points second derivatives.

Seven points third derivatives.

Seven points fourth derivatives.

Seven points fifth derivatives.

Eight points first derivatives.

Eight points second derivatives.

Eight points third derivatives.

Eight points fourth derivatives.

Eight points fifth derivatives.

Nine points first derivatives.

Nine points second derivatives.

Nine points third derivatives.

Nine points fourth derivatives.

Nine points fifth derivatives.

Ten points first derivatives.

Ten points second derivatives.

Ten points third derivatives.

Ten points fourth derivatives.

Eleven points first derivatives.

Eleven points second derivatives.

Eleven points third derivatives.

Eleven points fourth derivatives.

Eleven points fifth derivatives.