Some clarification questions; is the d in front of L, x, and y indicating that these values change?

Can someone please explain how they came up with the two equations from the pythagorean at the top? For the first one they seemed to have divided things by dx but then multiplied by dx squared, why? And same with the bottom one but with dy.

A question I have regarding the problem I was attempting was whats up with x=y squared. Can someone explain that; why not y=x squared, it seems vaguely familiar but I need a reminder of whats this called with an explanation.

Then what does dx/dy mean? Does that mean the derivative? Or to differentiate? I know how to find it if so by I just want to be sure thats why its equal to 2y.
My questions are abiuo the calculus involved not finding the centroid of the rod.
If im not mistaken the dy/dx is an alternative notation for the derivative. D, Dx and d/dx in front of say a function are differential operators indicating to differentiate. The dy/dx IS the derivative; not indicating to differentiate.