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 derivative is invariant under changes of chart and is thus well-defined for functions defined on a differentiable manifold. This notion is weaker than the classical directional derivative in general and equivalent to the latter for Lipschitz continuous functions. We characterize also the pairs of tangentially differentiable functions for which the chain rule holds.