If you have two derivatives, say $dx/dt$ and $dy/dt$, can you divide the two to get $dx/dy$?
I have looked around and the answer seems to be yes, but I have further questions. For example, what happens if $dy/dt=0$, so the denominator is $0$? I know that derivatives are limits, but we don't have as much to work with here, algebraically speaking. What if you divide two independent derivatives, say $(dw/dx)/(dy/dz)$ (including the case where they are all part of the same system, for example, if $dx/dy=1$)? How else could we algebraically manipulate derivatives (e.g. what happens if we multiply them? If $dx/dt$ is added to $dy/dt$, is the result $(dx+dy)/dt$? Essentially, how can derivatives be manipulated with and/or by each other?
I already tried searching around online with little success with my specific question.