We generalize the Newton-based extremum seeking algorithm, which, through measurements of an unknown map, maximizes the map's higher derivatives. Specifically, we propose a method for choosing the demodulation signals of a sinusoidally perturbed estimate, such that the extremum seeking algorithm maximizes the n th derivative only through measurements of the map. The Newton-based extremum seeking approach removes the dependence of the average convergence rate on the unknown Hessian of the higher derivative. This dependence is present in standard gradient-based extremum seeking while the average convergence rate of our parameter estimates is user-assignable. Our design stems from the existing multivariable Newton-based extremum seeking algorithm where a differential Riccati equation estimates the inverse Hessian of the function to be maximized. Algebraically computing a direct estimate of the inverse Hessian is susceptible to singularity, where-as employing the Riccati filter removes that potential. We prove local stability of the algorithm for general nonlinear maps via averaging theory.