In my classes, the derivative is usually defined as "the limit of the fractional incremental ratio". But I found out another way to define the derivative from an old book from Gardner & Thompson's "Calculus Made Easy". For example, if we have $f(x) = x^2$ and we want to calculate the derivative:
$ f(x) = x^2 $
So our y is
$ y = x^2 $
Right now we are only considering the increment of y and x so we can calculate that this way:
$ y+dy = (x+dx)^2 $ $ y+dy = x^2+dx^2+2xdx $
I can remove $ dx^2 $ because it's a very small quantity related to our magnitude.
The result is
$ y+dy = x^2+2xdx $
I subtract the original quantity $ y+dy-y=x^2+2xdx-x^2 $ $ dy=2xdx $ $ dy/dx=2x $
The derivative is equal to $2x$ and I calculate that without using any limits. So, my question is: the derivative is a real limit? What about the orders of magnitude? A limit is the representation of a value related to our magnitude?