Derivations with power values on multilinear polynomials

A polynomial f (X-1,X-2,..., X-n) is called multilinear if it is homogeneous and linear in every one of its variables. In the present paper our objective is to prove the following result: Let R be a prime K-algebra over a commutative ring K with unity and let f(X-1,X-2,..., X-n) be a multilinear polynomial over K. Suppose that d is a nonzero derivation on R such that df(x(1),x(2), ..., x(n))(s) = f(x(1), x(2), ... , x(n))(t) for all x(1), x(2), ... , x(n) is an element of R, where s, t are fixed positive integers. Then f (X-1,X-2,..., X-n) is central-valued on R. We also examine the case R which is a semiprime K-algebra.