We study certain properties of derivations on semiprime rings. The main purpose is to prove the following result: let R be a semiprime ring with center Z ( R ) , and let f , g be derivations of R such that f ( x ) x + x g ( x ) ∈ Z ( R ) for all x ∈ R , then f and g are central. As an application, we show that noncommutative semisimple Banach algebras do not admit nonzero linear derivations satisfying the above central property. We also show that every skew-centralizing derivation f of a semiprime ring R is skew-commuting.