  Recently, Gavazzo has developed a relational theory of symbolic manipulation,
that allows to study syntax-based rewriting systems without relying on specific
notions of syntax. This theory was obtained by extending the algebra of
relations with syntax-inspired operators. Within the algebras thus obtained, it
is possible to encode notions of parallel and full reduction for first-order
rewriting systems, as well as to prove nontrivial properties about them in an
algebraic and syntax-independent fashion. Sequential reduction, however, was
not explored, but it was conjectured that it could be studied through a
differential relational theory of rewriting. This manuscript proves the above
conjecture by defining differential algebras of term relations, viz. algebras
of term relations extended with novel operators inspired by the theory of
functor derivatives. We give a set of axioms and rules for such operators and
show that the resulting theory is expressive enough to define notions of
parallel, full, and sequential reduction. We prove fundamental results relating
all these notions in a purely algebraic and syntax-independent way, and
showcase the effectiveness of our theory by proving the soundness of a proof
technique for weak confluence akin to the so-called Critical Pair Lemma.
