I'm having trouble with understanding what exactly a differential really is.
For example, if we have the following function, $f(x,y)=x^2+xy+\frac{37}{x} +5$, does this differential, $df=\frac{\partial f}{\partial x}dx + \frac{\partial f}{\partial y}dy$, mean anything in relation to the original function? In this example $\frac{\partial f}{\partial x} = 2x+y-\frac{37}{x^2}$ and $\frac{\partial f}{\partial y} = x$.
Is the differential form a valid way of writing the original function?
It appears as though you lose information of any constants if you only have the differential form, so in what way(s) is a differential useful? The original function seems like it would be far more useful to use.
I should add that I only have a typical physics major's background in math: 3 semesters calculus, 1 semester ODE, 1 semester PDE, 1 semester linear algebra, 1 semester discrete math.