A simpler way is saying that a corner is a point of undefined curvature. Curvature can be defined a number of ways; you can determine its value with a rectangular, polar, or parametric function, or one of a few other ways.
Well, he said “polar differentiation,” so yes he is talking about dr/d(theta), which is equal to zero around the circle.
Of course. My bad. I had dy/dx stuck in my mind for some reason (so used to finding dy/dx for polar/parametric equations, I guess). That’s a reason, not an excuse.