Page:Calculus Made Easy.pdf/26

 that such quantities as $$x\cdot dx$$, or $$x^2dx$$, or $$a^xdx$$ are negligible. But $$dx\times dx$$ would be negligible, being a small quantity of the second order.

A very simple example will serve as illustration.

Let us think of $$x$$ as a quantity that can grow by a small amount so as to become $$x+dx$$, where $$dx$$ is the small increment added by growth. The square of this is $$x^2+2x\cdot dx+(dx)^2$$. The second term is not negligible because it is a first-order quantity; while the third term is of the second order of smallness, being a bit of, a bit of $$x^2$$. Thus if we took $$dx$$ to mean numerically, say, $$\tfrac{1}{60}$$ of $$x$$, then the second term would be $$\tfrac{2}{60}$$ of $$x^2$$, whereas the third term would be $$\tfrac{1}{3600}$$ of $$x^2$$. This last term is clearly less important than the second. But if we go further and take $$dx$$ to mean only $$\tfrac{1}{1000}$$ of $$x$$, then the second term will be $$\tfrac{2}{1000}$$ of $$x^2$$, while the third term will be only $$\tfrac{1}{1,000,000}$$ of $$x^2$$.



Geometrically this may be depicted as follows: Draw a square (Fig. 1) the side of which we will take to represent $$x$$. Now suppose the square to grow by having a bit $$dx$$ added to its size each