Geometric partial differentiability on manifolds: The tangential derivative and the chain rule

We define the tangential derivative, a notion of directional derivative which is invariant under diffeomorphisms. In particular this notion is independent of local charts and thus well-defined for functions defined on a differentiable manifold. This notion is stronger than the classical directional derivative and equivalent to the latter for Lipschitz continuous functions. We characterize also the pairs of tangentially differentiable functions for which the chain rule holds.