A least squares approach to differentiation (Q5919882)

The author considers a new approach of finding a slope of a function at a point \(c\). He starts by considering the usual derivative and the symmetric derivative at a point. There are taken several points on the given curve and found the least squares line of best fit, then take the limit of the slope of this line as the points converge to \((c,f(c))\). By considering \(2n+1\) symmetrically spaced data points and by taking the limit as \(n\) tends to infinity, integral formulas are obtained. Further, generalizations using Legendre polynomials are given. The author constructs a function which admits such a least square derivative at a point \(c\), but it has not symmetric derivative at that point.