Our main goal is to show that if there exist Jordan derivations D and G on a noncommutative (n + 1)!-torsion free prime ring R such that $$D(x)x^n-x^nG(x)in C(R)$$ for all $xin R$, then we have D=0 and G=0. We also prove that if there exists a derivation D on a noncommutative 2-torsion free prime ring R such that the mapping $chi$longrightarrow[aD($chi$), $chi$] is commuting on R, then we have either a = 0 or D = 0.