By employing the turbulent theoretical approach of Alber, Davidson and Roy and Bhakta the dynamical evolution of the wave spectrum of nonlinear wavetrains near marginal stability is derived, starting from a derivative nonlinear Schrödinger equation, extended with a quintic nonlinearity, called the EDNLS-equation. The stability of three homogeneous background spectral densities (the delta-, the normal- and the uniform distribution), is investigated in the linear regime. The most important findings in that respect as as follows: First, the delta-distribution instability criterion deviates by a numerical factor from the one presented in Flå and Wyller for the modulational instability in the EDNLS-equation. Secondly, the growth rate of this instability result in the limit of small, but finite spectral width limit is equivalent with the turbulence theory result based on the derivative nonlinear Schrödinger equation up to modulo translation in the central wave number.