Sensitivity analysis is an important tool in engineering with applications ranging from optimal design to computational fluid dynamics. There are many algorithms available for computing sensitivities. These range from automatic differentiation techniques to variational methods. In this paper we describe two variational methods for computing sensitivities. We use a 1D model problem as a platform for describing and comparing these methods. In particular, the regularity of the sensitivity equations obtained from each method is addressed. Numerical approximations to each sensitivity are calculated using finite element schemes, and numerical comparisons are presented.