  In this paper, a construction to imitate the Bernstein-Zelevinsky derivative
for automorphic representations on $GL_n(\mathbb{A})$ is introduced. We will
later consider the induced representation \[I(\tau_1,\tau_2;\underline{s}) =
\mathrm{Ind}_{P_{[n_1,n_2]}}^{G_n}(\Delta(\tau_1,n_1)|\cdot|^{s_1}\boxtimes
\Delta(\tau_2,n_2)|\cdot|^{s_2}).\] from the discrete spectrum representations
of $GL_n(\mathbb{A})$, and apply our method to study the degenerate Whittaker
coefficients of the Eisenstein series constructed from such a representation as
well as of its residues. This method can be used to reprove the results on the
Whittaker support of automorphic forms of such kind proven by D. Ginzburg, Y.
Cai and B. Liu. This method will also yield new results on the Eulerianity of
certain degenerate Whittaker coefficients.
