# 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.