Project I (Paul Falcone)

The Changing Concept of Change:The Derivative from Fermat to Weierstrass: Judith V. Grabiner in Mathematics Magazine Vol. 56 No. 5

# Article Summary

In this article, Grabiner describes the fascinating history and development of the derivative. The derivative's history is interesting because it is quite contrary to our intuition regarding the historical development of ideas and concepts. The author classifies four main stages in the derivative's history: its first usage by Fermat, its discovery by Newton and others, its development by Euler and Lagrange, and lastly, its rigorous definition given by Cauchy and Weierstrass. The author explains each of these stages in detail and the difficulties mathematicians have faced in working with the derivative.

Usage
The article begins by discussing the first real usage of the derivative by Pierre Fermat in his attempt to calculate the maximum and minimum values of a curve. To illustrate his method, Fermat, demonstrated that if a line is to be broken up into two segments, the maximum value of the product of their lengths occurs when the two lengths are equal. Fermat was also able to show that, given a nth degree polynomial, $\sum_n a_n x^n$, a maximum or minimum value occurs when $\sum_n n a_n x^{n-1}$=0. Fermat's method in demonstrating these facts was not, however, rigorous, although it foreshadowed and gave a basis for future mathematicians to build upon.

Discovery
The next period of the derivative's history was its "discovery" by Isaac Newton and Gottfried Leibniz. By discovery, the author means that they took previously known theorems and methods in dealing with tangent lines, maximum and minimum values of curves, rates of change, etc. and combined them in the branch of calculus now known as differential calculus. Both Newton and Leibniz were able to develop succinct and useful notation for differentiation, namely $\frac{dx}{dt}$, and they were both able to prove the Fundamental Theorem of Calculus, which showed that there is an inverse relationship between differentiation and integration. While Newton and Leibniz discovered the most important properties of the derivative, their analyis again lacked rigor, and there were still many properties of the derivative yet to be discovered. Newton and Leibniz lacked rigor in the same way Fermat lacked rigor: they never clearly defined what they meant by a differential quotient. Their vagueness in defining the differential quotient caused some scholars, such as George Berkeley, to criticize calculus as having an unstable foundation and it therefore being untrustworthy. The solidification of calculus's foundations would not come until much later, after Euler and Lagrange would discover many of the wonderful properties and applications of differentiation.

Development and Exploration
The study of differential equations was developed further in the 18th century primarily by Taylor series. Euler was able to demonstrate that Taylor series were effective at approximating the solution to many differential equations and in determining the maximum and minimum values of a function. However, he still was unable to put calculus on a stable and rigorous foundation. The French mathematician Joseph Lagrange was able to build upon Euler's work with Taylor series by attempting to simplify all of calculus to algebra; that is, he sought to reduce all functions, such as the sine and exponential functions, into infinite series, where one could simply use just algebra to deal with them. Lagrange sought to show that every analytic function could be written as a Taylor series as follows:

(1)
\begin{equation} $f(x+h)$=$f(x)$+$h p(x)$+$h^2$$q(x)+h^3$$r(x)$+... \end{equation}

Using this analysis, he was able to develop a new way to look at the derivative: as a function derived from another function (this is where we get the word derivative). One limitation to Lagrange's work, however, was that he assumed every differentiable function was the sum of a Taylor series, which is a false claim. For example, the function $f(x)=e^{-x^{-2}}$ has a Taylor series about x=0, but the function is not zero everywhere. The lack of a suitable for the derivative would motivate the next and final stage in the history of the derivative, as Cauchy and Weierstrass would provide the necessary rigor in defining differentiation.

Definition
Cauchy defined a derivative as follows:

(2)
\begin{align} $\lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$ \end{align}

However, unlike his predecessors, Cauchy defined limit in a completely different manner which allowed his definition of the derivative to remain rigorous. He never tried to determine whether or not the function ever reaches its limit because he realized such an abstraction was meaningless. His definition of a limit L of a function $f(x)$ at x=c is as follows: for each $\epsilon$ greater than zero, there exists a $\delta$ greater than zero, such that if $||{x-c}||$ is less than $\delta$, than$||{f(x)-L}||$ is less than $\epsilon$. Interestingly enough, his definitions of limits and derivatives are still taught today, as they give a definitive and eloquently algebraic meaning to the terms. His new definitions allowed the calculus to be viewed as a rigorous subject like geometry and allowed for rigorous proofs of all the major theorems in calculus. One limitation to Cauchy's work was that he failed to recognize the difference between convergence and uniform convergence, a detail that would be later worked out by the great German mathematician, Karl Weierstrass.

# Connection to Real Analysis and Broader Context

This article is an analysis on the history of differentiation, one of the key topics in real analysis. Cauchy's definition of a limit is absolutely crucial to analysis, without which analysis would quite simply fall apart. The history of the derivative helps explain why analysis is so important: without having the sure and rigorous foundation analysis provides, the calculus would become like another science and lose its absolute certainty which mathematicians hold so dear.
One of the author's objectives with this article is to show that there is more to mathematics than just the formulas and equations derived from definitions. There is a rich history behind every development, which is often ignored. The author believes is the historian's duty to capture the development of an idea as it actually occured, and it is to this end that this article was written.

For more information on this topic, see "History of the Infinitely Small and the Infinitely Large in Calculus" by Israel Kleiner in Educational Studies in Mathematics, Vol. 48, No. 2/3, Infinity: The Never-Ending Struggle. (2001), pp. 137-174. Stable URL: http://links.jstor.org/sici?sici=0013-1954%282001%2948%3A2%2F3%3C137%3AHOTISA%3E2.0.CO%3B2-A

Bibliography
The Changing Concept of Change:The Derivative from Fermat to Weierstrass
Judith V. Grabiner
Mathematics Magazine Vol. 56 No. 5. (Sept., 1983), pp. 195-206.
Stable URL: http://www.jstor.org/view/0025570x/di021131/02p02223/0?frame=frame&userID=86f09906@usma.edu/01c0a848750050a8c91&dpi=3&config=jstor