Q:

Derivative of volume of ball is surface area of sphere, but derivative of area of square is half its perimeter.

The derivative of the volume of a ball WRT radius is the surface area of the 2-sphere.
The derivative of the  area of a circle is WRT radius is the circumference. 
And yet, the derivative of the area of a square WRT length is only half its perimeter. Why?
I have knowledge of exterior calculus- that the exterior derivative can be interpreted as the boundary of a form. I believe that the phenomena of taking derivatives to recover the boundary of the manifold has something to do with this- but I do not know why this process only recovers half the perimeter of a square.

A:

Intuitively, if you increase the radius of a ball of radius $r$ by $dr$ the added volume is a shell with area the area of the sphere and thickness $dr$.  The new volume is $\frac 43 \pi (r+dr)^3-\frac 43\pi r^3 \approx 4\pi r^2 dr$.  Similarly if you expand a square of side $s$ by $ds$ the added area is the area between two squares.    The comparable variable to the radius for a square is half the side, so the area is $4(\frac s2)^2$.  The new area is $4(\frac s2 +d\frac s2)^2-4(\frac s2)^2\approx 4s\ d\frac s2$.  $4s$ is the perimeter of the square.

