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 ▫$R$▫ be a prime ring with ▫$\text{char}(R) = 0$▫ or ▫$4 < \text{char}(R)$▫, and let ▫$D \colon R \to R$▫ be an additive mapping satisfying either the relation ▫$D(x^3) = D(x^2)x + x^2D(x)$▫ or the relation ▫$D(x^3) = D(x)x^2 + xD(x^2)$▫ for all ▫$x \in R$▫. In both cases ▫$D$▫ is a derivation.
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