A typical example of a Cartan calculus is the Lie algebra representation of vector fields of a manifold on the derivation ring of the de Rham algebra. In this talk, a ”second stage” of the Cartan calculus is investigated. It is formulated as a homotopy Cartan calculus in the sense of Fiorenza and Kowalzig by using the André-Quillen cohomology and Hochschild homology of the de Rham algebra. Moreover, we also discuss a geometric description of the homotopy Cartan calculus in view of the isomorphism between the André-Quillen cohomology and the rational homotopy group of the monoid of self-homotopy equivalences. This talk is based on joint work with K. Kuribayashi, T. Naito and T. Yamaguchi.