Constants of formal derivatives of non-associative algebras, Taylor expansions and applications

We study unitary multigraded non-associative algebras R generated by an ordered set X over a field K of characteristic 0 such that the mappings d_k: x_l->delta_{kl}, x_k,x_l in X, can be extended to derivations of R. The class of these algebras is quite large and includes free associative and Jordan algebras, absolutely free (non-associative) algebras, relatively free algebras in varieties of algebras, universal enveloping algebras of multigraded Lie algebras, etc. There are Taylor-like formulas for R: Each element of R can be uniquely presented as a sum of elements of the form (...(r_0 x_{j_1})...x_{j_{n-1}}) x_{j_n}, where r_0 is a constant and j_1