How to graph this derivative?

Question

I asked a question about this problem previously but for this post I am asking a different question about the same problem.

Problem:

The volume of a cylinder equals cubic inches, where is a constant. Find the proportions of the cylinder that minimize the total surface area.

I know how to get the answer to this problem. What I have trouble with is visualizing what the graph of $\frac{dS}{dr}$ is if $S(r)$ is the total surface area as a function of the radius. The equation for $\frac{dS}{dr}$ is $\frac{dS}{dr}=\frac{4\pi r^3-2V}{r^2}$ and since $V=\pi r^2h$, shouldn't the total surface area be a function of both the radius and the height, so basically $S(r,h)$.

Answer 1

Here, Green axis : $h$, Red axis: $r$

Since, $$\cfrac{dS}{dr} = \cfrac{4 \pi r^3 - 2V}{r^2} = \cfrac{4 \pi r^3 - 2 \pi r^2 h}{r^2} = 2 \pi ( 2r - h) $$ If I define $z = \cfrac{dS}{dr}$ ,then $$ z = 2 \pi (2r-h) $$ is just equation of a plane.