I'm stumped with a question my boss posed to me today. He explained how since the area of a circle is Pi * r^2, its derivative is 2Pi * r, the same as it's circumference (due to the infinitesimal change in area equaling the circumference).

Now, take a square.

It's area is x^2

derivative = 2x

The perimeter of a square is 4x.

So why doesn't this Area Derivative/Perimeter relationship hold true for squares as it does circles?