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 Numerical differentiaon - Formulas of numerical differentiation 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)

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