In this paper, we provide an overview of state-of-the-art techniques that are being developed for efficient calculation of second and higher nuclear derivatives of quantum mechanical (QM) energy. Calculations of nuclear Hessians and anharmonic terms incur high costs and memory and scale poorly with system size. Three emerging classes of methods─machine learning (ML), automatic differentiation (AD), and matrix completion (MC)─have demonstrated promise in overcoming these challenges. We illustrate studies that employ unsupervised ML methods to reduce the need for multiple Hessian calculations in dynamics simulations and those that utilize supervised ML to construct approximate potential energy surfaces and estimate Hessians and anharmonic terms at reduced cost. By extension, if electronic structure operations could be written in a manner similar to functions underlying ML methods, rapid differentiation or AD routines can be employed to inexpensively calculate higher arbitrary-order derivatives. While ML approaches are typically black-box, we describe methods such as compressed sensing (CS) and MC, which explicitly leverage problem-specific mathematical properties of higher derivatives such as sparsity and low-rank, to complete higher derivative information using only a small, incomplete sample. The three classes of methods facilitate reliable predictions of observables ranging from infrared spectra to thermal conductivity and constitute a promising way forward in accurately capturing otherwise intractable higher-order responses of QM energy to nuclear perturbations.