Let R be a prime ring with extended centroid C, λ a nonzero left ideal of R and f (X 1, . . . , X t ) a nonzero multilinear polynomial over C. Suppose that d and δ are derivations of R such that $$d(f(x_{1},\ldots,x_{t}))f(x_{1},\ldots,x_{t})-f(x_{1},\ldots,x_{t})\delta(f(x_{1},\ldots,x_{t}))\in C$$ for all $${x_1,\ldots,x_t\in\lambda}$$ . Then either d = 0 and λ δ(λ) = 0 or λ C = RCe for some idempotent e in the socle of RC and one of the following holds: (1)
f (X 1, . . . , X t ) is central-valued on eRCe;
(2)
λ(d + δ)(λ) = 0 and f (X 1, . . . , X t )2 is central-valued on eRCe;
(3)
char R = 2 and eRCe satisfies st 4(X 1, X 2, X 3, X 4), the standard polynomial identity of degree 4.