Optimal points for numerical differentiation

Newn-pointrth derivative Lagrangian numerical differentiation formulas employ the best irregular locations of points, A. From the standpoint of highest degree accuracy for derivatives at a singlefixed point (nth degree accuracy proven to be the highestexactly attainable for anyr). B. From the Tschebyscheff standpoint of minimal largest |remainder| over an argument range. In B the dominant term in the remainder is minimal for arguments at the zeros ofrth order integrals of Tschebyscheff polynomials specialized by addition of suitable (r−1)th degree polynomials chosen to produce real, distinct locations of points within or fairly close to the range of optimization. First and second derivative formulas up to nine-point, are obtained with remainder estimates.