In this thesis a new class of rank‑1 update formulas is investigated. It is applied to approximate Jacobians in quasi-Newton methods for nonlinear equations. The update formulas make explicite use of adjoint and partly direct derivative information. These derivatives can be efficiently evaluated, for example, by methods of Automatic Differentiation. A particular feature of the update formulas is that they combine the fixed scale least change property of Broyden’s update with the heredity property of the SR1 update. This allows to prove rapid local superlinear convergence. Moreover, in combination with line search, global convergence results are also established. The new quasi-Newton methods are applied to various test problems and numerical result compare them to other well established methods.