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: