This might be a trivial question (I assume it is at least) but I was wondering if someone could help me out with it. I have an equation given as follows: $${\partial_{t}v + \frac{\sigma^{2}}{2}\frac{\partial_{yy}v}{(\partial_{y}v)^{2}} = \frac{\sigma^{2}}{2}} \tag 1$$
Now, I am told that by taking a variable $w$ to be the inverse of $v$ we now obtain the equation:
$$\partial_{t}w + \frac{\sigma^2}{2}\partial_{xx}w + \frac{\sigma^{2}}{2}\partial_{x}w = 0 \tag 2$$
with
$$\partial_{x}w = \frac{1}{\partial_{y}v} = \frac{1}{s}$$
My question is, how do we obtain the equation $\partial_{t}w + \frac{\sigma^2}{2}\partial_{xx}w + \frac{\sigma^{2}}{2}\partial_{x}w = 0$ ? I assume that for the variable $w$ we must take $w = \frac{1}{v}$ to have the inverse for $v$, but beyond this I have no idea what sort of arrangement of equation $(1)$ will yield equation $(2)$. I think it should be fairly trivial - the researcher I'm working with mentioned that it was obvious/straightforward so I'm sure I'm missing something stupid (Unfortunately I can't talk to them since they're in another country now...). If anyone can help me out with this I'd really appreciate it, thanks in advance.