We consider P₂(Rᵈ), the set of Borel probability measures of finite second moments on Rᵈ, which we endow with the Wasserstein metric W₂. It is well–known that P₂(Rᵈ), W₂), is isometric to a quotient space of the Hilbert space H of square-integrable random variables on (0, 1)ᵈ. We elucidate the connection between various notions of differentiability in the Wasserstein space: some have been introduced intrinsically (in the Wasserstein space, by using typical objects from the theory of Optimal Transport). Another notion is extrinsic and arises from the identification of the Wasserstein space with the Hilbert space of square-integrable random variables on a non-atomic probability space.