[bmitc]

What is the meaning then?

dy = f'(x)dx is just a definition for notional convenience, primarily employed when doing u or u-v substitution. My point is that dx in single variable calculus is notation. It is not an intrinsic object. dx is an intrinsic object as a differential form on a smooth manifold. Of course, the real line R is a 1-manifold, so dx does have that meaning, but you need to understand what a differential form is to know that.

One doesn't necessarily need the full generality of smooth manifolds though. Harold Edwards' Advanced Calculus: A Differential Forms Approach and Advanced Calculus: A Geometric View teach differential forms for Euclidean manifolds.

[Koshkin]

I think I get your point, but at the same time I disagree.

1) The differential of a function (at a point), dy, is not notation, it is a concept.

2) The differential of the function y = x, dx, is not, then, a notation, either; and, since the derivative is 1, dx = 1 Δx = Δx = x - x0.

3) You can argue, of course, that using dx instead of Δx in dy = f'(x)dx is "notation," but I think the above shows that it is more than that.

[bmitc]

The second book is by James Callahan. I accidentally left that off.