How are infinitesimals of polar coordinates "treated"?
E.g.
what does
$$\frac{d}{dx}=\bigg( \frac{dr}{dx} \frac{d}{dr}\bigg) + \bigg( \frac{d \theta}{dx} \frac{d}{d \theta}\bigg)$$
mean?
Particularly how does one "evaluate" $\frac{dr}{dx}$ or $\frac{d}{dr}$ etc.?
If $(x,y)$ denote Cartesian coordinates, then you have transformation laws $r=r(x,y)$ and $\theta=\theta(x,y)$, which can be found inverting the equations $x=r\cos \theta$ and $y = r\sin \theta$. If you have a function $f=f(r,\theta)$ this is also a function of $(x,y)$, because $f(r,\theta)=f(r(x,y),\theta(x,y))$, so it makes sense to compute the total derivative of $f$ with respect to $x$. By the chain rule you get $$\frac{\mathrm{d}f}{\mathrm{d}x} = \frac{\partial f}{\partial r}\frac{\partial r}{\partial x}+\frac{\partial f}{\partial \theta}\frac{\partial \theta}{\partial x},$$ from which you get the identity for differential operators: $$\frac{\mathrm{d}}{\mathrm{d}x} = \frac{\partial }{\partial r}\frac{\partial r}{\partial x}+\frac{\partial }{\partial \theta}\frac{\partial \theta}{\partial x}.$$