Based on a Gelfand triple constructed as a nuclear rigging of a Boson Fock space, we study the white noise operator theory including the analytic characterization of operator symbols and Fock expansion, and we introduce the new notion of quantum white noise derivatives, annihilation and creation derivatives, of white noise operators.  Then we study basic properties of the quantum white noise derivatives and then all Wick derivations on the space of white noise operators are characterized in terms of annihilation and creation derivatives.