Calculating partial derivatives for this particular function

Question

For $s,r>0$, we have a function $V(s,r)= (gs^2+r(1-g^2-2\ln s))/2$.

The paper that I am reading says that $V_s=gs$. However, shouldn't it be $gs-r/s$?

This is a highly cited paper, and hence I'm unable to understand how it could have such a simple error

Answer 1

Quoting the paper:

$V(s,r)= (gs^2+r(1-g^2-2\ln s))/2$, where $g(r,s)=h^{-1}(r/s^2)$ and [...] $(h^{-1})'(t)=\frac1{2th^{-1}(t)-1}.$

Therefore, $$g_s=\frac{-2r/s^3}{2(r/s^2)g-1}=\frac{r/s}{s^2/2-rg}$$ and $$\begin{align}V_s&=gs-r/s+(s^2/2-rg)g_s\\ &=gs-r/s+r/s\\ &=gs.\end{align}$$