Sure it does. There is no need to know about smooth manifolds or differential forms to understand the differential of a function of one variable at a point and the meaning of dy = f’(x)dx.
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.
Any introductory calculus book worth the paper it’s printed on would gladly tell you that the differential of the function y = x at a point x0 is nothing more than x - x0 and that you do not have to think about it as something that is “infinitely small” or anything equally mysterious. (Some would even go as far as saying that “the differential of a function of one variable is a linear map of the increment of the argument.”) So, with dx = x - x0, you can do with it anything you want, even divide by it (assuming that dx stays non-zero).
Sure it does. There is no need to know about smooth manifolds or differential forms to understand the differential of a function of one variable at a point and the meaning of dy = f’(x)dx.