Derivations on commutative regular algebras

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