A Leibniz Notation for Automatic Differentiation

Abstract Notwithstanding the superiority of the Leibniz notation for differential calculus, the dot-and-bar notation predominantly used by the Automatic Differenti-ation community is resolutely Newtonian. In this paper we extend the Leibnitz no-tation to include the reverse (or adjoint) mode of Automatic Differentiation, and use it to demonstrate the stepwise numerical equivalence of the three approaches using the reverse mode to obtain second order derivatives, namely forward-over-reverse, reverse-over-forward, and reverse-over-reverse