It is commonly assumed that calculating third order information is too expensive for most applications. But we show that third-order directional derivatives can be calculated at a cost proportional to that of a state-of-the-art method for calculating the Hessian matrix (second-order derivatives). We do this by first presenting a simple procedure for designing high order reverse methods and applying it to deduce several methods including a reverse method that calculates the third-order directional derivatives. We have implemented this method taking into account symmetry and sparsity, and successfully calculated this derivative for functions with a million variables. These results indicate that the use of third order information in a general nonlinear solver, such as Halley–Chebyshev methods, could be a practical alternative to Newton’s method. Furthermore, high-order sensitivity information is used in methods for robust aerodynamic design. An efficient high-order differentiation tool could facilitate the use of similar methods in the design of other mechanical structures.