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 ▫$n \ge 3$▫ be some fixed integer, let ▫$R$▫ be a prime ring with ▫$char(R) > 4n-8$▫ and let ▫$D \colon R \to R$▫ 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.
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