Let \(\mathcal {R}\) be a n!-torsion free semiprime ring and \(\Delta :\mathcal {R}^n\rightarrow \mathcal {R}\) be a symmetric n-derivation with the trace \(D:\mathcal {R}\rightarrow \mathcal {R}\) . In this article we will study the following identities: (1) \([D(s),D(t)]=\pm [s^n,t^n]\) ;
(2) \([D(s),t^n]-[s^n,D(t)]=0\) ;
(3) \(D([s,t])=[D(s),t^n]\) ;
(4) \( D(s)\circ D(t)=\pm s^n\circ t^n\) ;
(5) \([D(s),D(t)]=\pm s^n\circ t^n\) ;
(6) \( D(s)\circ D(t)=\pm [s^n,t^n]\) ;
(7) \([D(s),D(t)]=\pm s^n t^n\) ;
(8) \([D(s),D(t)]=\pm t^n s^n\) ;
(9) \(D(s)D(t)=\pm [s^n,t^n]\) ;
for all \(s,t\in \mathcal {R}\) .
