THE DIFFERENTIAL OF THE SUM OF SEVERAL FUNCTIONS IS THE SUM OF THEIR DIFFERENTIALS. CONSEQUENCES OF THIS PRINCIPLE. DIFFERENTIALS OF IMAGINARY FUNCTIONS.

Abstract

In previous lectures, we have shown how we form the derivatives and the differentials of functions of a single variable. We now add new developments to the study that we have made to this subject. Let x always be the independent variable and $\varDelta x=\alpha h=\alpha dx$ an infinitely small increment attributed to this variable. If we denote by $s, u,$ $v, w, \dots$ several functions of x,  and by $\varDelta s, \varDelta u,$ $\varDelta v, \varDelta w, \dots$ the simultaneous increments that they receive while we allow x to grow by $\varDelta x,$ the differentials $ds, du,$ $dv, dw, \dots$ will be, according to their own definitions, respectively, equal to the limits of the ratios

$$\begin{aligned} \frac{\varDelta s}{\alpha }, \ \ \frac{\varDelta u}{\alpha }, \ \ \frac{\varDelta v}{\alpha }, \ \ \frac{\varDelta w}{\alpha }, \ \ \dots . \end{aligned}$$