Derivations on commutative regular algebras

For a regular (in the sense of von Neumann) algebra $\mathcal A$ over an algebraically closed field of characteristic $0$, we describe the linear space $\mathcal D(\mathcal A)$ of all derivations on $\mathcal A$. The description is obtained in terms of algebraically independent elements of $\mathcal A$. In particular, we estimate the dimension of the space $\mathcal D(\mathcal A)$, where $\mathcal A=S[0,1]$ is the algebra of measurable functions on $[0,1]$.