This document introduces a notion of functional differences in which the difference of a function f with respect to a function h is that function g that describes how the value of f changes when its argument is altered by h: f(h x) = g(f x). It also introduce the inverse operation of functional integration and derive useful properties of both operations. The result is a calculus that facilitates derivation and reasoning about recursive programs. This is illustrated in a number of simple examples. The author presents preliminary results pertaining to fixed differences, that is, functional differences that do not depend on the value of the argument x.