The volume of a sphere of radius *r* is .

The surface area of a sphere of radius *r* is .

Why is the formula for the surface area *S* equal to the derivative of
the formula for the volume? Because when *r* is increased very
slightly by an amount ,
the volume *V* of the sphere increases very slightly by an amount
approximately equal to (a ``shell'' around the sphere of
area *S* and width ). So ,
and in the limit *dV*/*dr*=*S*.

Wed Apr 21 08:17:28 EDT 1999