In which frame are $d$ and $\partial$ rigourous?

Question

I know $\frac{dy}{dx}$ may be a shortcut of $\lim_{h \to 0} \frac{y(x+h)-y(x)}{h}$, which is totally rigourous, but it loses that sense if I write $dx=f\cdot dy$.

How could that become rigourous again and in what frame?