Identities with Skew Derivations

We solve affirmatively a problem, raised by Kharchenko, on identities with compositions of skew derivations: We define the notion of trivial identities with compositions of skew derivations, which is unique in a certain sense. It is proved that if a prime ring R satisfies a nontrivial identity with compositions of skew derivations, then R also satisfies a generalized polynomial identity (without skew derivations). We actually work in a more general context, in which higher (skew) derivations in the literature known to the author are all covered.