Proof of polar dy/dx?

i.e. $\frac{\mathrm{dy}}{\mathrm{dx}} = \setminus \frac{\dot{y}}{\setminus} \dot{x} = \frac{r ' \setminus \sin \setminus \theta + r \setminus \cos \setminus \theta}{r ' \setminus \cos \setminus \theta - r \setminus \sin \setminus \theta}$ My assumption is that it's the product rule, given that $y = r \setminus \sin \setminus \theta$ and $x = r \setminus \cos \setminus \theta$

$\setminus \dot{y} = \frac{d}{d \setminus \theta} \left(r \setminus \sin \setminus \theta\right) = r ' \setminus \sin \setminus \theta + r \setminus \cos \setminus \theta$
$\setminus \dot{x} = \frac{d}{d \setminus \theta} \left(r \setminus \cos \setminus \theta\right) = r ' \setminus \cos \setminus \theta + r \left(- \setminus \sin \setminus \theta\right)$