Derivative of area =circumference

[HyperbolicMan]

For a circle, the derivative of the formula for area is the formula for circumference:

d/dr([pi*r^2)=2*pi*r.

Similarly, the derivative of the formula for the volume of a sphere is equal to the formula for surface area: d/dr(4/3*pi*r^3)=4*pi*r^2.

I'm positive that these are consequences of Stokes's (Green's) and Divergence theorems, but I don't know how to show this. Could someone please explain? Thanks.

 

[mathman]

Looking at it as an integration exercise makes things clearer. Add up circular lamina of length 2πr and width dr to get area. Similarly shells of area 4πr² and width dr to get volume.