I read this question the other day and it got me thinking: the area of a circle is $\pi r^2$, which differentiates to $2 \pi r$, which is just the perimeter of the circle.
Why doesn't the same thing happen for squares?
If we start with the area formula for squares, $l^2$, this differentiates to $2l$ which is sort of right but only half the perimeter. I asked my calculus teacher and he couldn't tell me why. Can anyone explain???