I was looking through a paper that described a simple payoff function where there is an outcome variable $Y$ that depends on some causation variable $X$ and the payoff is given as some function of the two as $\pi(X,Y)$. Now, when they took the derivative w.r.t to $X$, they ended up with this equation:
$\dfrac{d\pi(X,Y)}{dX} = \dfrac{\partial\pi}{\partial{X}}(Y) + \dfrac{\partial\pi}{\partial{Y}}\dfrac{\partial{Y}}{\partial{X}}$
I don't understand how this equation makes sense for a generic two-variable function. I suppose that since $Y$ depends on $X$ in some (unknown) way, this problem could be simplified as $Y=f(X)$ but even then, this doesn't seem to hold for some example payoff functions. For example, if $\pi(X,Y) = X+Y$, the $Y$ term in the derivative equation would be removed.
I feel like I'm missing something obvious here, just not sure what it is.
By the chain rule, $\frac{d\pi(X,Y)}{dX}=\frac{\partial\pi}{\partial X}\frac{\partial X}{\partial X}+\frac{\partial\pi}{\partial Y}\frac{\partial Y}{\partial X}.$ $\frac{\partial X}{\partial X}$ is, of course, $1$, leading to the r.h.s. in the formula you cite. The author appears to be emphasizing that the $\pi_X$ term is a function of $Y$.