Sensitivity analysis (SA) concerns the quantification of changes in Partial Differential Equations (PDEs) solution due to perturbations in the model input. SA has many applications, among which uncertainty quantification, quick evaluation of close solutions, and optimization based on descent methods. Standard SA techniques for PDEs, such as the continuous sensitivity equation method, rely on the differentiation of the state variable. However, if the governing equations are hyperbolic PDEs, the state can exhibit discontinuities yielding Dirac delta functions in the sensitivity. The aim of this work is to modify the sensitivity equations to obtain a solution without delta functions. This is motivated by several reasons: firstly, a delta function cannot be seized numerically, leading to an incorrect solution for the sensitivity in the neighborhood of the state discontinuity; secondly, for some applications like optimization and evaluation of close solutions, the peaks appearing in the numerical solution of original sensitivity equations make sensitivities unusable. We propose a two-steps procedure: (i) definition of a correction term added to the right-hand side of the sensitivity equations; (ii) definition of a shock detector ensuring that the correction term is added only where needed. We show how this procedure can be applied to the Euler barotropic system with two different finite-volume formulations, based either on an exact Riemann solver or the approximate Roe solver.