I have a related rates problem. The problem asks for the rate at which the height of water in an inverted right circular cone is falling, given the rate at which water is leaking from the cone. I don't think the values of the variables are necessary to my question, so I omit them here. The formula for the volume of a right circular cone is $V = \frac {\pi \ r^2 h}{3}$.
My question: Does the following correctly express the derivative of volume with respect to time?
$$ \frac {dV}{dt} = \frac {2 \pi r}{3} \frac {dr}{dt} \frac {dh}{dt}$$
I am quite sure that this is not correct, but I cannot articulate why. (FYI, I realize this type of problem might be solved by using similar triangles to eliminate $r$ from the equation or by using partial derivatives. I just can't articulate why the formulation above is not correct.)