A finite generalization of differentials?

So basically, in trying to make sense of a certain math aspect of a thermodynamic problem (how to manipulate differentials) I end up reading this http://www.tau.ac.il/~corry/teaching/toldot/download/Bos1974.pdf

After reading it, I now have some preliminary idea on manipulating differentials (at least for those back in the days of Lebniz) but it also raise some interesting thoughts

1. Are there mathematical entities which has the following property?

a. Similar to differentials, they are in different orders such that if say an operator $a$ can turn x into an entity such that

$a(x)$ is infinitely smaller than $x$ and

$a^2(x)$ is infinitely infinitely smaller than $x$

b. Multiplication by a FINITE value $b$ can bring the $a$'s to other orders of infinity

e.g. $$b \times a(x)$$ has the same order as $x$ and $$b^2 \times a^3(x)$$ has the same order as $a(x)$.

Phrasing in this way, differentials are a special case of these class of objects when setting $b$ tends to infinity

In short, what is the field of maths that study objects that are sort of like a generalisation of differentials in that

objects can be made infinitely larger or smaller than another by multiplying by a certain finite number $b$?

I also noticed the idea of different scales reminds of the Hausdorff dimensions of fractals http://davis.wpi.edu/~matt/courses/fractals/intro.html Which basically showing how things from different scales are related to each other,

1. is the mathematical class of objects I am trying to seek related to fractals?
• Look up non-standard analysis (or the hyperreal numbers). They act like this. – Milo Brandt Nov 27 '14 at 5:13