The author proves two theorems about derivations which concern annihilators of elements. For both, let \(D\) be a nonzero derivation of the ring \(R\). Theorem 1. Let \(R\) be semi-prime and also \((n-1)!\) torsion free. If \(tD(x)^ n=0\) for some \(t\in R\) and all \(x\in R\), then \(tD(x)=0=D(x)t\), and when \(R\) is a prime ring, \(t=0\). This result is used to prove: Theorem 2. If \(R\) is prime with \(\text{char}(R)\neq 2\), and if for all \(x\in R\), \(f(x)D(x)=0=D(x)f(x)\) for an additive mapping \(f\) of \(R\), then \(f=0\).