We elucidate the connection between different notions of differentiability in $\mathcal P_2(\mathbb R^d)$: some have been introduced intrinsically by Ambrosio--Gigli--Savar\'e, the other notion due to Lions, is extrinsic and arises from the identification of $\mathcal P_2(\mathbb R^d)$ with the Hilbert space of square-integrable random variables. We mention potential applications such as uniqueness of viscosity solutions for Hamilton-Jacobi equations in $\mathcal P_2(\mathbb R^d)$, the latter not known to satisfy the Radon--Nikodym property. (This talk is based on a work in progress with A Tudorascu).