Derivation of the second- and higher-order virial coefficients for models of the gaseous state is demonstrated by employing a direct differential method and subsequent term-by-term comparison to power series expansions. This communication demonstrates the application of this technique to van der Waals representations of virial coefficients. Consistency of the technique is demonstrated by application to the Dieterici gas state model for the determination of its second and third virial coefficients. Conclusions are then drawn regarding the similarities and differences of these model equations based on the dependence of their virial coefficients on real gas correction parameters.