A LEAST SQUARES APPROACH TO DIFFERENTIATION

We consider an approach to differentiation that involves least squares lines of best fit rather than the traditional secant lines and use elementary techniques to show how this leads to the Lanczos derivative. A number of examples are presented to illustrate this concept and to show that the product, quotient, and chain rules fail for the Lanczos derivative. Several results giving conditions for which these rules do hold are discussed and proved. A brief introduction to higher order Lanczos derivatives is included.