I am trying to understand some text from the Sampling: Design and Analysis textbook on stratum sampling.
To provide some background information, the variance of the mean is $$V(\bar{y}_{str}) = \sum_{h=1}^H(1 - \frac{n_h}{N_h})N_h^2\frac{S_h^2}{n_h}$$
where $N_h$ is the sampling units in stratum h, and $n_h$ is the set of units actually used in the simple random sample for stratum h.
$S_h^2$ is the population variance in stratum h, and is equal to $\sum_{j=1}^{N_H}\frac{(y_{hj}-\bar{y}_{hU})^2}{N_h-1}$
Now, to what I don't understand:
I particularly don't understand how using calculus, we can prove that the optimal allocation has $n_h$ proportional to $$\frac{N_hS_h}{\sqrt{c_h}}$$