  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.
