Why the surface area of only the sphere is the derivative of the volume, when this is not the case for other single variable shapes?

Answer 1

A counter question:

Why would we expect ANY shape to have that property?

I've never seen this before---but I still think I can explain it.....

A sphere has the property of complete symmetry and uniformity with respect to the center. Any cut that goes through the center results in the same shape. There is no other solid shape that has this property.

If you take a very small slice, you can see that the VOLUME of the sphere is aggregation of all the slices, whereas the surface area is only the aggregation of the surface area.

Answer 2

That is an excellent philosophical question. It's a hard one to answer, too. You may find my answer less than satisfying, but this is true:

A sphere is a continuous shape.

Other common 3-d shapes (e.g. cube, cone, prism and all that) are not. They have discontinuities (i.e., edges)

Derivatives are generally undefined at a discontinuity of the function. And there is your answer.