Derivative Relationships in a Circle and in a Sphere
by David Pleacher
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:
