  Covariant Lyapunov vectors or CLVs span the expanding and contracting
directions of perturbations along trajectories in a chaotic dynamical system.
Due to efficient algorithms to compute them that only utilize trajectory
information, they have been widely applied across scientific disciplines,
principally for sensitivity analysis and predictions under uncertainty. In this
paper, we develop a numerical method to compute the directional derivatives of
CLVs along their own directions. Similar to the computation of CLVs, the
present method for their derivatives is iterative and analogously uses the
second-order derivative of the chaotic map along trajectories, in addition to
the Jacobian. We validate the new method on a super-contracting Smale-Williams
Solenoid attractor. We also demonstrate the algorithm on several other examples
including smoothly perturbed Arnold Cat maps, and the Lorenz attractor,
obtaining visualizations of the curvature of each attractor. Furthermore, we
reveal a fundamental connection of the CLV self-derivatives with a statistical
linear response formula.
