We study the stabilizability of a linear controllable system using state derivative feedback control. As a special feature the stabilized system may be fragile, in the sense that arbitrarily small modeling and implementation errors may destroy the asymptotic stability. First, we discuss the pole placement problem and illustrate the fragility of stability with examples of a different nature. We also define a notion of stability, called $p$-stability, which explicitly takes into account the effect of small modeling and implementation errors. Next, we investigate the effect on the fragility of including a low-pass filter in the control loop. Finally, we completely characterize the stabilizability and $p$-stabilizability of linear controllable systems using state derivative feedback. In the stabilizability characterization the odd number limitation, well known in the context of the stabilization of unstable periodic orbits using Pyragas-type time-delayed feedback, plays a crucial role.