  In recent years, algebraic studies of the differential calculus and integral
calculus in the forms of differential algebra and Rota-Baxter algebra have been
merged together to reflect the close relationship between the two calculi
through the First Fundamental Theorem of Calculus. In this paper we study this
relationship from a categorical point of view in the context of distributive
laws which can be tracked back to the distributive law of multiplication over
addition. The monad giving Rota-Baxter algebras and the comonad giving
differential algebras are constructed. Then a mixed distributive law of the
monad over the comonad is established. As a consequence, we obtain monads and
comonads giving the composite structures of differential and Rota-Baxter
algebras.
