How to graph this derivative?

Problem:

The volume of a cylinder equals 𝑉 cubic inches, where 𝑉 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 $$S(r,h)$$.

• You can use a $3$ dimensional surface with axes $S,r,h$. Why not ask the same question for $V$ itself which depends on both $r$ and $h$? – Peter Foreman Jun 30 '19 at 16:16
• What you want to graph $S(r,h)$ or dS/dr ? – Ajay Mishra Jun 30 '19 at 16:29
• @AjayMishra $\frac{dS}{dr}$, but how do I graph this if I don't know what $V$ is? – user532874 Jun 30 '19 at 16:39
• @user532874 see my answer. – Ajay Mishra Jun 30 '19 at 16:45

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.
• How can $\frac{dS}{dr}$ have just $dr$ on the bottom when both $r$ and $h$ are inputs? – user532874 Jun 30 '19 at 16:49
• Oh, it is just $\cfrac{ \partial S}{ \partial r}$ . Are you familiar with partial derivatives? – Ajay Mishra Jun 30 '19 at 16:53