This paper presents a two-phase approach for the calculation of second-order derivatives for fixed-point iterations. Such an approach computes derivatives by auxiliary fixed-point iterations performed after the one that computes the fixed point itself. Compared with a straightforward application of Algorithmic Differentiation (AD) to fixed-point iterations, this saves run-time and tape memory. The convergence rate of the new fixed-point iteration is similar to the rate of the original fixed-point iteration. We discuss the iteration loops for the derivatives, their convergence behavior, and appropriate termination criteria. Two numerical examples confirm the predicted convergence rates and show the performance benefit