After working with instantaneous rates of change and related
rates problems, students often notice that taking the derivative
of the area of a circle yields the circumference and
taking the derivative of the volume of a sphere gives the formula
for the surface area. There is a reason for that -- it is
not just coincidence.

(1) To show that the derivative of the area of a circle
        equals the circumference:

       
        This definition represents the difference of the
        areas of a circle of (r + h) radius and a circle of r
        radius. As h approaches 0, that ring (shown as red in
        the diagram below) becomes the circumference.

           
       

(2) To show that the derivative of the volume of a sphere
        equals the surface area: