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

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.