The Arithmetic of Differentiation

This report describes automatic differentiation, which is neither symbolic nor approximate, for single functions of one real variable. The rules of evaluation and differentiation are combined into an ordered-pair arithmetic similar to complex arithmetic, but slightly simpler. Evaluation of the formula for a function in this arithmetic yields both the values of the function and its derivative, without a formula for the derivative of the function, and without numerical approximation, since this arithmetic is baed on the well-known rules for differentiation. The properties of this arithmetic are examined, and illustrated by simple examples. Subroutines are given for differentiation arithmetic both on a hand-held programmable calculator, and in the microcomputer language Pascal-SC. An application of this arithmetic to the solution of equations by Newton's method is given, using a Pascal-SC program. (Author)