An Introduction to a Rigorous DeÖnition of Derivative 1 Introduction The concept of a derivative evolved over a great deal of time, originally driven by problems in physics and geometry. Mathematicians found methods that worked, but justiÖcations were not always very convincing by modern standards. A co-inventors of Calculus, Isaac Newton XXXXXXXXXX), is generally regarded as one of the most ináuential scientists in human history. Newton used the term áuxion for what we now call a derivative. He developed most of his methods around 1665, but did not publish them immediately. Later Newton wrote articles and books about his áuxion methods, which ináuenced many mathematicians across Europe. Newton and áuxions. The modern deÖnition of a function had not yet been created when Newton developed his áuxion theory. The context for Newtonís methods of áuxions is a particle tracing out a curve in the x y plane. The x and y coordinates of the moving particle are áuents or áowing quantities. The horizontal and vertical velocities are the áuxions of x and y, respectively, associated with the áux of time. In the excerpt below from [N], Newton is considering the curve y = x n and wants to Önd the áuxion of y: XXXXXXXXXXLet the quantity x áow uniformly, and let it be proposed to Önd the áuxion of x n : In the same time that the quantity x; by áowing, becomes x+o; the quantity x n will become (x + o) n ; that is, ... x n + noxn1 + n 2 n 2 ooxn2 + &c: And the augments o and noxn1 + n 2 n 2 ooxn2 + &c: are to one another as 1 and nxn1 + n 2 n 2 oxn2 + &c: Now let these augments vanish, and their ultimate ratio will be 1 to nxn1 : XXXXXXXXXXExercise 1 Write out the algebraic details of Newtonís áuxion method for n = 3 using modern algebraic notation. Exercise 2 Convert Newtonís argument that (x n ) 0 = nxn1 to one with modern limit notation for the case where n is a positive integer. You may use modern li It is important to remember that when Newton developed his áuxion method, there was no theory of mathematical limits. Critics of Newtonís áuxion method were not happy with having augment o be a seemingly nonzero value at the beginning of the method, and then have the augment o ìvanishî at the end of the argument. Indeed, the philosopher and theologian George Berkeley XXXXXXXXXXwrote a 1734 paper [B] attacking Newtonís methods. Berkeley rejected: ì...the fallacious way of proceeding to a certain Point on the Supposition of an Increment, and then at once shifting your Supposition to that of no Increment ... Since if this second Supposition had been made before the common Division by o, all had vanished at once, and you must have got nothing by your Supposition. Whereas by this ArtiÖce of Örst dividing, and then changing your Supposition, you retain 1 and nxn1 . But, notwithstanding all this address to cover it, the fallacy is still the same. And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither Önite Quantities nor Quantities inÖnitely small, nor yet nothing. May we not call them the ghosts of departed quantities?î Some historians argue that Berkeley was attacking the Calculus itself, while others argue that he was instead claiming that Calculus was no more logically rigorous than theology. Certainly, many a Calculus I student has had the same feeling that Berkeley had about the limiting process! As we shall see in Section 2, by the early 1800ís the mathematical community was close to putting limits and derivatives on a Örmer mathematical foundation. LíHÙpital on the di§erential of a Product. While Newton was working on his áuxion methods, G. Leibniz XXXXXXXXXXwas independently developing calculus in Germany during the same time period. Many of Leibnizís ideas appeared in the 1696 book Analyse des inÖnitment petits [LH] by G. LíHÙpital XXXXXXXXXXA key idea for Leibniz was the di§erential. Here is an excerpt from LíHÙpitalís book XXXXXXXXXXDeÖnition II. The inÖnitely small part whereby a variable quan