calculus - What is the meaning of a differential in terms of an exact differential?

As I understand it a differential is an outdated concept from the time of Liebniz which was used to define derivatives and integrals before limits came along. As such $dy$ or $dx$ don't really have any meaning on their own. I have seen in multiple places that the idea of thinking of a derivative as a ratio of two infinitesimal change while intuitive is wrong. I understand this, and besides I am not even really sure if there is a rigorous way of saying when a quantity is infinitesimal.

Now on the other hand, it have read that you can define these differentials as actual quantities that are approximations in the change of a function. For example for a function of one real variable the differential is the function $df$ of two independent real variables $x$ and $Δx$ given by:

$df(x,Δx)=f'(x)Δx$

How this then reduces to

$df = f'(x)dx$

and again what $dx$ means I dont understand.

It seems to me that it is simply a linear approximation for the function at a point $x$. However there's no mention of how large or small $dx$ must be, it seems to be just as ill defined as before and I have still found other places referring to it as an infinitesimal even when it has been redefined as here.

Anyway ignoring this, I can see how this could then be extended to functions of more than one independent variable

$y = f(x_1,....,x_n)$

$dy = $$\frac{df}{dx_1}dx_1\ +\ .... \ +\frac{df}{dx_n}dx_n\ $

However then the notion of exact and inexact differentials are brought up. This seems like its unrelated but that raise the question of what a differential means in this case.

All this comes from a course I am taking in Thermal Physics.]2

If anyone can enlighten me as to what the concept of differentials means or perhaps direct me towards a book or website where I can study it myself I would be very grateful.

An explanation of Schwarz' Theorem in this context would be great too.