Let R be a prime ring of characteristic different from 2, with extended centroid C and with maximal right ring of quotients \(Q_{mr}(R)\) . Suppose that \(f:R\rightarrow Q_{mr}(R)\) and \(d:R\rightarrow Q_{mr}(R)\) are additive maps satisfying \(f(x^{2})=f(x)x+xd(x)\) for all \(x\in R\) . Then there exist \(a\in Q_{mr}(R)\) , a derivation \(\delta :R\rightarrow Q_{mr}(R)\) and an additive map \(\mu :R\rightarrow C\) such that \(f(x)=ax+\delta (x)+\mu (x)\) and \(d(x)=\delta (x)-\mu (x)\) for all \(x\in R\) , where \(\mu (x^{2})=0\) for all \(x\in R\) . This result is a natural generalization of the notions of generalized derivations and (generalized) Jordan derivations of prime rings.