We prove that a semiprime ring R must be commutative if it admits a derivation d such that (i) x y + d ( x y ) = y x + d ( y x ) for all x , y in R , or (ii) x y − d ( x y ) = y x − d ( y x ) for all x , y in R . In the event that R is prime, (i) or (ii) need only be assumed for all x , y in some nonzero ideal of R .