Consider a spherical cap, for which the base radius is $a$ and the height is $h$. Then, the surface area and volume is (these equations can be found on Wolfram Mathworld)
$A(a,h) = \pi(a^2 +h^2)$,
$V(a,h) = \frac{\pi h}{6}(3a^2 +h^2)$.
Taking the partial differential with respect to $a,h$ would give
$dA = 2\pi a \ da + 2\pi h \ dh$
$dV = \pi ha \ da + \frac{\pi}{2}(a^2+h^2) \ dh$.
My question is: how would one evaluate $dA/dV$ when both $A$ and $V$ are described by more than one variable? For a sphere or a hemisphere, one can use the chain rule
$\frac{dA}{dV} = \frac{dA/dr}{dV/dr}$
as both $A$ and $V$ can be written as a function of $r \equiv 2h/(a^2+h^2)$, and one could easily find $dA/dV = 2/r$.
You could write both $a$ and $h$ as functions of a single parameter $t$, and replace $da$ and $dh$ with $\frac{da}{dt}$ and $\frac{dh}{dt}$. Then you're effectively saying that $a$ and $h$, both being dependent on $t$, are not independent as they are in $V(a, h)$ and $A(a, h)$. But this is okay as finding the total derivative implicitly assumes such a dependence.
Alternatively, you can explicitly write $h$ as a function of $a$ or vice versa.