On concepts of directional differentiability

Various definitions of directional derivatives in topological vector spaces are compared. Directional derivatives in the sense of Gâteaux, Fréchet, and Hadamard are singled out from the general framework of -directional differentiability. It is pointed out that, in the case of finite-dimensional spaces and locally Lipschitz mappings, all these concepts of directional differentiability are equivalent. The chain rule for directional derivatives of a composite mapping is discussed.