izabo said:
df = {\frac{\partial f}{\partial x}} dx + {\frac{\partial f}{\partial y}} dy
that: dx \neq \partial x.
If you want to make sense of this for a function ##f:\mathbb R^2\to\mathbb R## (without using definitions from differential geometry), then you should define ##df:\mathbb R^4\to\mathbb R## by
$$df(x,y,z,w)=D_1f(x,y)z+D_2f(x,y)w$$ for all ##x,y,z,w\in\mathbb R##. This ensures that for all ##x,y,dx,dy\in\mathbb R##, we have
$$df(x,y,dx,dy)=D_1f(x,y)dx+D_2f(x,y)dy.$$ What you wrote can be considered a sloppy notation for this result.
This is a way to make sense of df, dx and dy. I don't think I have ever seen anyone try to make sense of ##\partial x##. It's just a small part of the notation ##\partial f/\partial x##, which means ##D_1f##. I find the notation ##\partial f/\partial x## misleading, since it hides the fact that ##D_1f## (the partial derivative of f with respect to the first variable slot) is the same function no matter what symbols we usually use to represent the numbers we plug into ##f##.
Note that with this definition, there's no need for dx and dy to be "infinitesimal". What does that even mean? I know that there's a definition of that term, but I haven't studied it, and I'm pretty sure that almost no one of the authors of physics books who use that term have either. If you see the term "infinitesimal" in a physics book, you should assume that it has nothing to do with infinitesimals. You should interpret it as a code that let's you know that the next thing that follows is a first-order approximation. For example, to say that for infinitesimal x, we have ##e^x=1+x##, is to say (in a weird and confusing way) that there's a function ##R:\mathbb R\to\mathbb R## such that ##e^x=1+x+R(x)## for all ##x\in\mathbb R##, and ##R(x)/x\to x## as ##x\to 0##.
df(x,y,z,w) is a first-order approximation of the difference f(x+z,y+w)-f(x,y). So if we want to be weird and confusing, we can say that for infinitesimal z and w, we have f(x+z,y+w)-f(x,y) = df(x,y,z,w). If we want to cause some additional confusion, we can use the notations dx and dy for the real numbers z and w, and then drop the (x,y,z,w) from df(x,y,z,w), and the (x,y) from ##D_1f(x,y)## and ##D_2f(x,y)##. Then we would be saying that for infinitesimal dx and dy, we have
$$f(x+dx,y+dy)-f(x,y)=df=D_1f dy+ D_2f dx =\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy.$$