# Servois' 1814 Essay on a New Method of Exposition of the Principles of Differential Calculus, with an English Translation - Lagrange's fonction derivee

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What follows is a summary of Lagrange’s method of series expansion, which is contained in [Lagrange 1797, pp. 1-15]. Many of the details can be found in his Théorie des fonctions analytiques [Grabiner 1981, 1990] and [Katz 2009, pp. 633-636].

Lagrange began by taking $f(x)$ to be an arbitrary function of $x$. Then, if $h$ is an indeterminate quantity, he supposed he could form an infinite series in terms of $h$, $f(x + h) = f(x) + ph + qh^{2} + rh^3 + \cdots,\quad\quad (4)$ where $p, q, r, \ldots$ are new functions of $x$, independent of $h$, and are derived from the original function $f(x)$.

To find the exact terms of the power series, Lagrange wrote series (4) in the following form, $f(x + h) = f(x) + h\left[P(x, h)\right],$ where $P(x, h)$ represents the difference quotient, $P(x, h) = \frac{f(x + h) - f(x)}{h}.$ Lagrange argued that it is possible to separate from $P$, the part $p$, which does not vanish when $h = 0$. Therefore, $p(x) = P(x, 0)$ and $Q(x, h) = \frac{P(x, h) - p(x)}{h},$ or $P = p + hQ$. Thus, $f(x + h) = f(x) + ph + h^{2}Q$. Continuing similarly, we can let $Q = q + hR$, where $q(x) = Q(x, 0)$. Then $f(x + h) = f(x) + ph + qh^2 + h^3R$. The continuation of this process yields expansion (4).

The coefficients $p, q, r, \ldots$ are derived from $f(x)$ and Lagrange called them fonctions dérivées (this is where our modern term “derivative” comes from). Lagrange used the notation $f^{\prime}(x)$ for $p$ and then investigated the relationship among $p$, $q$, $r$, …. By considering the expansion of $f(x + h + i)$ in two different ways, where $i$ is another indeterminate increment, he showed that $p = f^{\prime}(x)$, $2q = p^{\prime}$, $3r = q^{\prime}$, …. Expressing all of these derived functions in terms of $f(x)$ gives series (4) the familiar form of the Taylor series: $f(x) + f^{\prime}(x)h + \frac{f^{\prime\prime}(x)}{2!}h^2 + \ldots.$

By taking $h$ to be sufficiently small, but still finite, Lagrange argued he could control the error in any approximate value of $f(x + h)$ based on finitely many terms in the series (4). In particular, he showed that $f(x + h) = f(x) + f^{\prime}(x)h + \frac{f^{\prime\prime}(x)}{2!}h^2 + \ldots + \frac{f^{(n)}(x)}{n!}h^{n} + \frac{f^{(n+1)}(x+i)}{(n+1)!}h^{n+1},$ for some value of $i$ satisfying $0 < i < h$. The term $\frac{f^{(n+1)}(x+i)}{(n+1)!}h^{n+1}$ is therefore called the Lagrange Remainder Term for the Taylor series.

Robert E. Bradley (Adelphi University) and Salvatore J. Petrilli, Jr. (Adelphi University), "Servois' 1814 Essay on a New Method of Exposition of the Principles of Differential Calculus, with an English Translation - Lagrange's [i]fonction derivee[/i]," Convergence (January 2011), DOI:10.4169/loci003597