I have always had an issue with how one seems to be able to multiply through by say $dx$, which shouldn't be really allowed as $dy/dx$ is really $d/dx$ operating on y. I would like to rigioursly prove that starting from
$$ dF=\frac{\partial F}{\partial x}dx + \frac{\partial F}{\partial y}dy $$ we can get to $$ \frac{\partial F}{\partial t}=\frac{\partial F}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial F}{\partial y}\frac{\partial y}{\partial t}. $$ My approach: Integrate both sides along some path to get $$F=\int_{\gamma}^{} dF= \int_{\gamma_{x} } \frac{\partial F}{\partial x}dx + \int_{\gamma_{y} } \frac{\partial F}{\partial y}dy .$$ Then $$\frac{\partial F}{\partial \theta}= \lim_{\delta \theta \to 0 } (?) .$$
I can't seem to formulate taking the limit of this expression (how do I treat the differentials?). If anybody can help me understand this I would greatly appreciate it. Also if anybody has a rigorous proof somewhere I would really appreciate a reference.