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

• Dennis M. Cates
Chapter

## 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}

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## Authors and Affiliations

1. 1.Sun CityUSA