Adjoint-based first-order perturbation theory is applied again to boundary perturbation problems. Rahnema developed a perturbation estimate that gives an accurate first-order approximation of a flux or reaction rate within a radioactive system when the boundary is perturbed. When the response of interest is the flux or leakage current on the boundary, the Roussopoulos perturbation estimate has long been used. The Rahnema and Roussopoulos estimates differ in one term. This paper shows that the Rahnema and Roussopoulos estimates can be derived consistently, using different responses, from a single variational functional (due to Gheorghiu and Rahnema), resolving any apparent contradiction. In analytic test problems, Rahnema’s estimate and the Roussopoulos estimate produce exact first derivatives of the response of interest when appropriately applied. A realistic, nonanalytic test problem is also presented.