A result in the spirit of Herstein theorem

Sa？etak A classical result of Herstein asserts that any Jordan derivation on a prime ring of characteristic different from two is a derivation. It is our aim in this paper to prove the following result, which is in the spirit of Herstein's theorem. Let \(n≥ 3\) be some fixed integer, let R be a prime ring with \(char(R)> 4n-8\) and let D:R → R be an additive mapping satisfying either the relation \(D(x^n)=D(x^{n-1})x+x^{n-1}D(x)\) or the relation \(D(x^n)=D(x)x^{n-1}+xD(x^{n-1})\) for all \(x \in R\). In both cases D is a derivation