The commutative/classical Cartan calculus amonts to a Lie type action of vector fields of a smooth manifold on its de Rham complex of differential forms. Its noncommutative analogue is expected to take the form of a homotopy Lie type action of the Hochschild cochain complex of a (homotopy) associative algebra on its the Hochschild chain complex compatible with Connes' boundary map. Such a structure implies a noncommutative chain-level version of the Gauss-Manin connexion. In this talk, I will explain how the operadic calculus allows one to solve this problem and I will provide fully explicit formulas in terms of an operad introduced by Kontsevich—Soibelman.