Il teorema fondamentale del calcolo

The fundamental theorem of calculus began with the seventeenth-century discovery that finding tangents to curves and finding areas under curves are inverse processes. It took abstract form in the statement that differentiation and integration of functions are inverse operations, then continued to evolve as the notions of differentiation, integration, and function were clarified and extended in various ways. One of the most remarkable features of this story is the irritant role of the infinitesimal, a concept that seemed for a long time to be absurd, yet indispensable. The search for relief from the irritation of infinitesimals led not only to clarification of the concept of function, but also to a precise concept of number itself. Thus, there is reason to claim that the fundamental theorem of calculus is the most fundamental theorem in the history of mathematics, because it provoked an overhaul of mathematics from top to bottom.