Derivations in prime rings

Let $R$ be a ring and $d \ne 0$ a derivation of $R$ such that $d({x^n}) = 0$, $n = n(x) \geqslant 1$, for all $x \in R$. It is shown that if $R$ is primitive then $R$ is an infinite field of characteristic $p > 0$ and $p\vert n(x)$ if $d(x) \ne 0$. Moreover, if $R$ is prime and the set of integers $n(x)$ is bounded, the same conclusion holds.