Formulas of numerical differentiation

1. On a basis of Newton’s first interpolating formula

For finding of the first and second derivatives of function , given in equidistant points ( i = 0, 1, 2, …, n ) of segment [ a , b ] by values , it is approximately replaced with Newton’s interpolating polynomial constructed for system of nodes [1]:

(5)

Removing brackets and considering, that

we’ll receive:

.        (6)

Similarly, considering

we’ll receive:

.       (7)

In the same way if necessary, it is possible to calculate any order derivative of function. We’ll notice, that at calculation of derivatives in fixed point х as it is necessary to take the nearest tabular value of argument.

It is possible to deduce also formulas of numerical differentiation based on Newton’s second interpolating formula [1].

2. On a basis of Newton’s second interpolating formula

Let – system of equidistant points with step and corresponding values of given function. Putting and replacing the function by Stirling’s interpolating polynomial, we’ll receive:

(8)

where for brevity of record the following designations are entered:

and so on.

From (8) in view of that , follows:

(9)

. (10)