The scientific motivation for this paper is lack of clarity about the interdependency of primary and initial secondary equilibrium paths in the frame of sensitivity analysis of elastic structures. The investigation of this interdependency comprises of the following four cases: (1) nonlinear primary path, nonlinear stability problem, (2) linear primary path, nonlinear stability problem, (3) nonlinear primary path, linear stability problem, and (4) linear primary path, linear stability problem. The consistently linearized eigenproblem is used for differentiation of two classes of nonlinear stability problems with markedly different characteristics of both the prebuckling and the postbuckling behavior. For one of them, e.g. zerostiffness postbuckling is impossible. For the other one, which is restricted to a prebuckling regime with axial deformations only, sensitivity analysis of the initial postbuckling behavior either exhibits its continuous improvement or its continuous deterioration, depending on whether the bifurcation point diverges from or converges to the snap-through point. In other words, a monotonic variation of the design parameter cannot result in a non-monotonic change of the initial postbuckling behavior. The practical motivation for this work is to explore the mechanical reasons for qualitatively different modes of transition from imperfection sensitivity to insensitivity in the course of sensitivity analysis for the purpose of improving the postbuckling behavior of structures by means of minor design changes. Results from a numerical investigation corroborate the theoretical findings.