The authors prove that: (i) The only Jordan derivations \(D\) and \(G\) on a non-commutative \((n+1)!\) torsion free prime ring \(R\) such that \(D(x)x^n-x^nG(x)\) is in the center of \(R\) for all \(x\in R\) are \(D=0\) and \(G=0\); (ii) The only derivation \(D\) on a non-commutative 2-torsion free prime ring \(R\) such that the mapping \(x\rightarrow aD(x)x-xaD(X)\), \(a\in R\) nonzero, is commuting on \(R\) is \(D=0\).