"The Multiple Faces of the Derivative

In a typical first course of Calculus students are introduced to the idea of derivative as a point-value operation many pages, definitions, theorems and exercises prior to the treatment of the idea of integral. This usage is at least 1000 years in "the opposite direction" to the historical development of Mathematics. One may wonder why such giants as Arquimedes and Galileo to name just two of the most powerful mathematical minds ever to exist, have not caught up with the idea of derivative during that long dormancy. Of course the obvious explanation for it is that the concept of derivative is not simple and those who think otherwise are outliers or plain liars. The goal of the present article is to pause a look into this concept by mixing history, motivation and examples which are not usually found in common textbooks.