I'm blanking out here. If I had dx/dt, dy/dt, how would I find dy/dx? Is it as simple as just dividing the two out?
It's a combination of the fact that $\displaystyle \frac{dt}{dx} = \frac{1}{\frac{dx}{dt}}$ and the chain rule, namely that $\displaystyle \frac{dy}{dx} = \frac{dy}{dt}\,\frac{dt}{dx}$.
So yes, you DO "divide it out", but that's not really the right language to use.