The shape gradients of an instability''s growth rate and frequency are derived for an unstable mode calculated from a global stability analysis. These are calculated and interpreted physically for the flow around a cylinder at a Reynolds number of 50. This is a well-known canonical flow, which is often used to discover fundamental behaviour in bluff body flows and to test new numerical techniques. This paper shows that shape deformations affect hydrodynamic oscillations mainly through their influence on the steady base-flow, rather than through their influence on the unsteady feedback mechanism. Deformations that strongly affect the base-flow are shown to strongly affect the frequency and growth rate, as expected. In addition, subtle deformations at the rear of the cylinder are shown to exploit small base-flow changes that have a disproportionately large influence on the growth rate. The physical mechanism behind this is shown to be similar to the well-known phenomenon of ?base bleed?. The method presented in this paper is general and versatile. It provides engineers with gradient information in order to optimize designs systematically. In addition, it provides physical insight, which enables intuitive design changes that would be outside the range of an optimization algorithm or existing geometric parametrization.