Nevertheless, for simple computations this specific algebra indeed boils down to "applying the chain rule to the AST" (as well as the diff rules for all primitive functions). In forward mode, the only difference to symbolic differentiation is that common terms are reused (items in the AST, treating the AST as DAG), so the expression terms do not explode in size, and the computation is very efficient.

In reverse mode (and all other, more complicated modes), the chain rule still plays a central role, but is applied in a different way. Here, saying the "chain rule is applied to the AST" might still be technically correct, but would be very misleading, as the whole control flow is different, not to mention memory usage.