The shape derivability analysis of the flow of a viscous and incompressible fluid surrounding a rigid body B is considered. The novelty being in the choice of the boundary condition on the body B where we impose the so-called Navier boundary condition. Well-posedness of the time-dependent Navier-Stokes equations with mixed boundary conditions, of Navier and Dirichlet type, is established under regularity and smallness assumptions. After proving the shape differentiability of the state system, we compute the first order necessary optimality condition associated to drag shape minimization problem.