Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/163

Rh As the differential of a function is in general also a function of the independent variable, we may deal with its differential. Consider the function

$$\scriptstyle{d(dy)}$$ is called the second differential of $$\scriptstyle{y}$$ (or of the function) and is denoted by the symbol

Similarly, the third differential of $$\scriptstyle{y}$$, $$\scriptstyle{d[d(dy)]}$$, is writtenand so on, to the nth differential of $$\scriptstyle{y}$$,

Since $$\scriptstyle{dx}$$, the differential of the independent variable, is independent of $$\scriptstyle{x}$$ (see footnote, p. 131), it must be treated as a constant when differentiating with respect to $$\scriptstyle{x}$$. Bearing this in mind, we get very simple relations between successive differentials and successive derivatives. For and since dx is regarded as a constant.

Also, and in general

Dividing both sides of each expression by the power of $$\scriptstyle{dx}$$ occurring on the right, we get our ordinary derivative notation

Powers of an infinitesimal are called infinitesimals of a higher order. More generally, if for the infinitesimals $$\scriptstyle{\alpha}$$ and $$\scriptstyle{\beta}$$,then $$\scriptstyle{\beta}$$ is said to be an infinitesimal of a higher order than $$\scriptstyle{\alpha}$$.