In this paper we prove the following result. Let ▫$m \ge 0$▫ and ▫$n\ge 0$▫ be integers with ▫$m+n \ne 0$▫ and let ▫$R$▫ be a prime ring with ▫$char(R)=0$▫ or ▫$m+n+1 \le char(R) \ne 2$▫. Suppose ...there exists a nonzero additive mapping ▫$D:R \to R$▫ satisfying the relation ▫$D(x^{m+n+1}) = (m+n+1)x^m D(x)x^n$▫ for all ▫$x \in R$▫. In this case ▫$D$▫ is a derivation and ▫$R$▫ is commutative.
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