Let $R$ be a prime ring with center ${\cal Z}$ and let $f(X_1,...,X_n)$ be a multilinear polynomial which is not central-valued on $R$. Suppose that $d$ and $\delta $ are derivations on $R$ such that $d(f(x_1,..., x_n))f(x_1,..., x_n)-f(x_1,...,x_n)$ $\delta (f(x_1,...,x_n)) \in {\cal Z}$ for all $x_1,...,x_n$ in some nonzero ideal of $R$. Then either $d =\delta = 0$ or $\delta =-d$ and $f(X_1,...,X_n)^2$ is central-valued on $R$, except when char $R = 2$ and $R$ satisfies the standard identity $s_4$ in 4 variables.