This article deals with problems arising in the sensitivity analysis for fluid-structure interaction systems. Our model consists of a fluid described by the incompressible Navier-Stokes equations interacting with a solid under large deformations. We obtain a linearized problem which allow us to compute the derivative of the state variable with respect to a given boundary parameter. We use a particular deflinition of the first-order correction for the perturbed state and consider a weak Arbitrary Euler-Lagrang- e formulation for the coupled system.