What is the analogue of surface area and volume for a hypersphere? What about for n-spheres? Can you determine an explicit formula for those in terms of n? That seems to be the easiest way to demonstrate that one is the derivative of the other.
I've noticed that the circumference of a circle is the derivative of its area. I've also noticed that the surface area of a sphere is the derivative of its volume. Why is this true? Can this correlation be extrapolated to hyperspheres and all n-spheres?