Historical considerations for the teaching of the derivative

Some methods that were developed throughout history to find derivatives from Apolonio de Perga to Cauchy are presented, then itineraries are proposed for the introduction of the derivative in the curriculum: addressing positions of the theory of Action, Process, Object and Scheme (APOE) and the tools proposed by the known as the Onto-Semiotic Approach (OSA), for elements of the derivative before the limit and limits before derivatives. From these, semiotic representations are proposed that allow to approach the derivative (numerical, algebraic, formal, geometric, infinitesimal, local, variational and computational related approximation). Finding that the concept as taught today does not coincide with the development of its genesis and that a way of understanding the concept of derivatives from the use of treatments and conversions of registers of representation.