When I see the words "instantaneous rate of change", I have to assume that you're in some stage of pre-calculus in your math class.
The instantaneous rate of change of a function is just its first derivative.
We have the function
V(r) = 3 π r²
and we need its first derivative with respect to ' r '. That shouldn't be
too hard, because the ' 3 π ' is nothing but constants.
Watch me while I do it slowly for you:
-- The derivative of ' r² ' with respect to ' r ' is ' 2r '.
-- The derivative of V(r) with respect to ' r ' is (3 π) times the derivative of ' r² '.
-- The derivative of V(r) with respect to ' r ' is (3 π) times (2r) = 6 π r .
The value of the derivative when r=3 is (6 π 3) = 18π = about 56.5 feet³/foot .
