Why is the surface area of 3-D objects equal to the derivative of their Volume formulas?

Answer 1

To answer this question it may make more sense to look at it from the other way around, "Why is the volume the antiderivative (integral) of the surface area?"

An integral gives the area under a curve. Any shape, whether it be 2-D or 3-D, is just a curve. A 3-D shape is a curve revolved around an axis. So a way to look at the surface area is that is the curve revolved around an axis that generates this shape. The integral would give the area under said curve and when it is revolved it becomes the volume.

This means that the volume is the integral of surface area and surface area is the derivative of volume.