The volume v of a right circular cylinder of height 3 feet and radius r feet is v=v(r)=3(pi)(r^2). find the instantaneous rate of change of the volume with respect to the radius at r=3.

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 .