Derivative of area =circumference

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SUMMARY

The derivative of the area formula for a circle, represented as d/dr(πr²), results in the circumference formula, 2πr. Additionally, the derivative of the volume formula for a sphere, d/dr(4/3πr³), yields the surface area formula, 4πr². These relationships are direct applications of calculus and are linked to Stokes's and the Divergence theorems. Viewing these derivatives through the lens of integration, particularly by summing circular lamina and shells, clarifies the underlying concepts.

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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.
 
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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.
 

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