The following technical report is available from http://aib.informatik.rwth-aachen.de: Algorithmic Differentiation of Numerical Methods: Tangent-Linear and Adjoint Solvers for Systems of Nonlinear Equations Uwe Naumann, Johannes Lotz, Klaus Leppkes, and Markus Towara AIB 2012-15 We consider the Algorithmic Differentiation (also know as Automatic Differentiation; AD) of numerical simulation programs which contain calls to solvers for parameterized systems of n nonlinear equations. The local computational overhead as well as the additional memory requirement for the computation of directional derivatives or adjoints of the solution vector with respect to the parameters can quickly become prohibitive for large values of n. Both can be reduced drastically by the semi-discrete and continuous approaches to differentiation of the underlying numerical method to be discussed in this paper.