  Many problems in Physics and Chemistry are formulated as the minimization of
a functional. Therefore, methods for solving these problems typically require
differentiating maps whose input and/or output are functions -- commonly
referred to as variational differentiation. Such maps are not addressed at the
mathematical level by the chain rule, which underlies modern symbolic and
algorithmic differentiation (AD) systems. Although there are algorithmic
solutions such as tracing and reverse accumulation, they do not provide human
readability and introduce strict programming constraints that bottleneck
performance, especially in high-performance computing (HPC) environments. In
this manuscript, we propose a new computer theoretic model of differentiation
by combining the pullback of the $\mathbf{B}$ and $\mathbf{C}$ combinators from
the combinatory logic. Unlike frameworks based on the chain rule, this model
differentiates a minimal complete basis for the space of computable functions.
Consequently, the model is capable of analytic backpropagation and variational
differentiation while supporting complex numbers. To demonstrate the generality
of this approach we build a system named CombDiff, which can differentiate
nontrivial variational problems such as Hartree-Fock (HF) theory and multilayer
perceptrons.
