Sensitivity Analysis on ODE system

Hi everyone,
I am trying to use automatic differentiation in julia to compute the Jacobian of my outputs wrt my design variables for local sensitivity, but my ODEs are nearly coupled as for example du[3]=A*du[2]+B u[1]. This lead to very high values (~e23) for Jacobian elements. So far I am using an implicit solver (FBDF) to solve my ODE system. Do you have any idea on how can I perform a sensitivity analysis for a system like this in julia?