GENERALIZED TOPOLOGICAL OPERATOR ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-OPERATOR) THEORY IN GENERALIZED TOPOLOGICAL SPACES ($\mathcal{T}_{\mathfrak{g}}$-SPACES): PART IV. GENERALIZED DERIVED ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-DERIVED) AND GENERALIZED CODERIVED ($\operatorname{\mathfrak{g}-\mathfrak{T}_{\mathfrak{g}}}$-CODERIVED) OPERATORS