Leibniz formula in algebraic $K$-theory

The paper can be considered as an addendum to a paper of Thomason and Throbaugh where $K$-theory of algebraic varieties is equipped with relative $K$-groups. It is proved that this enriched $K$-theory satisfies the Panin–Smirnov axioms for ring cohomology theories of algebraic varieties. In particular it is proved that the Leibnitz formula, describing an interaction between a multiplication and a differential, holds in this case. A language of symmetric spectra and of monoidal model categories is used.