A new method for evaluating generalized derivatives in nonsmooth problems is reviewed. Lexicographic directional (LD-)derivatives are a recently developed tool in nonsmooth analysis for evaluating generalized derivative elements in a tractable and robust way. Applicable to problems in both steady-state and dynamic settings, LD-derivatives exhibit a number of advantages over current theory and algorithms. As highlighted in this article, the LD-derivative approach now admits a suitable theory for inverse and implicit functions, nonsmooth dynamical systems and optimization problems, among others. Moreover, this technique includes an extension of the standard vector forward mode of automatic differentiation (AD) and acts as the natural extension of classical calculus results to the nonsmooth case in many ways. The theory of LD-derivatives is placed in the context of state-of-the-art methods in nonsmooth analysis, with an application in multistream heat exchanger modelling and design used to illustrate the usefulness of the approach.