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.