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In Section 18, Servois develops the rules of the differential calculus. He begins by listing the *elementary simple* functions of a single variable \(x\): \[x^m, \quad a^x, \quad \ln x, \quad \sin x, \quad \cos x,\] and the *elementary composed* functions: \[\varphi (x) \cdot \psi (x), \quad (\varphi (x))^m, \quad a^{\varphi (x)}, \quad \ln (\varphi (x)), \quad \sin (\varphi (x)), \quad \cos (\varphi (x)).\] Before Servois can start developing the rules of the differential calculus he must first establish a well-known theorem, the chain rule; see Servois’ equation (81). Using the chain rule he finds the differentials of exponential functions and then derives the power rule and product rule. From a modern point-of-view, we would say that Servois used logarithmic differentiation in the derivation of the power and product rules. This is a natural progression for Servois, because he has observed that his differential and the natural logarithm behave similarly.

Servois then turns to differentiating the trigonometric functions \(\sin x\) and \(\cos x\). Unfortunately, the series for the differential is not easy to apply to trigonometric functions. Therefore, Servois introduced a new function, which he used to evaluate trigonometric differentials, \[F(x) = \frac{\cos (\alpha x) + \sqrt{-1}\sin (\alpha x)}{\cos^{x} (\alpha)}, \quad\quad(6)\] where \(\alpha\) is a constant and the increment in the variable \(x\) is 1. This is the only place in the “Essay" where Servois uses complex numbers. He assumes the reader is familiar with their properties. He doesn’t give any motivation for definition (6), nor does he make any mention of De Moivre’s formula in the numerator. Throughout his papers Servois was always careful to provide credit to discoveries made by other mathematicians. Because Servois did not attribute (6) to anyone else, we suspect that it is original to him. Therefore, Petrilli [2009, p. 91] called this “Servois’ Function.”

Using the angle sum formula for sine and cosine, Servois shows that \[\Delta^{m} F(x) = F(x) \cdot \left(\sqrt{-1}\tan \alpha\right)^{m},\] which gives him an expression for \({\mbox d} F(x)\) in equation (90). Comparing this to the expression \({\mbox d} F(x)\) given by the product rule and differentiating \(\cos^2 (\alpha x) + \sin^2 (\alpha x) = 1\) implicitly to write \({\mbox d} \cos (\alpha x)\) in terms of \({\mbox d} \sin (\alpha x)\) allows him to give a simple expression for \({\mbox d} \sin (\alpha x)\) in terms of a constant \(A\), which depends only on \(\alpha\). Although \(A\) would appear to have an imaginary component, Servois shows that it is in fact real and proves that \[{\mbox d} \sin x = \cos x \, {\mbox d} x \quad \mbox{and} \quad {\mbox d} \cos x = -\sin x \, {\mbox d} x.\]

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 - The Rules for Differential Calculus," *Loci* (January 2011), DOI:10.4169/loci003597