When thinking of polar coordinates, like you said we find dy/dx through manipulation of parametric forms of x and y. But what does this mean? Just like with a normal derivative in Cartesian coordinates, dy/dx at x=a represents the slope at that point. If the slope becomes infinite for a point or even for multiple points, that means that the graph is like you said going straight up towards infinity at those points. But that doesn't mean that it will reach infinity immediately. Once the slope changes to something finite again, the graph will move a different direction.
I would remember these two distinctions.
1)f(a) is the position of "a" on the actual graph.
2)f'(a) represents how fast the graph is increasing/decreasing at that point, but only at that point - not necessarily at points close to "a"