Reverse Faà di Bruno's Formula for Cartesian Reverse Differential Categories

Reverse differentiation is an essential operation for automatic differentiation.Cartesian reverse differential categories axiomatize reverse differentiation in a categorical framework, where one of the primary axioms is the reverse chain rule, which is the formula that expresses the reverse derivative of a composition.Here, we present the reverse differential analogue of Fa di Bruno's Formula, which gives a higher-order reverse chain rule in a Cartesian reverse differential category.To properly do so, we also define partial reverse derivatives and higher-order reverse derivatives in a Cartesian reverse differential category.