I'm reading a game-theory related paper*, and I'm not following the derivation of some property of the best-response functions.
Suppose I have two players $1$ and $2$, whose strategies are continuous levels of effort $x_1$ and $x_2$ respectively. The first order condition is given by for a Nash equilibrium strategy is given just standardly by $argmax_{x_1}(\pi_1)$:
$$\frac{\partial \pi_i}{\partial x_1}=\frac{\alpha\sigma h'(x_1)h(x_2)}{(\sigma h(x_1)+h(x_2))^2}-1=0$$
Let $x_1=r_1(x_2)$ denote player $1$'s reaction function. Since it is derived from player $1$'s first-order condition (above), we obtain it's derivative by differentiating along [the above FOC]:
$$\frac{dr_1(x_2)}{dx_2}=\frac{h'(x_1)h'(x_2)(\sigma h(x_1)-h(x_2))}{h(x_1)[h''(x_2)(\sigma > h(x_1)+h(x_2)-2h'(x_2))^2]}$$
Note that this is not the derivative of the first equation w.r.t. $x_2$, which simply is
$$\frac{\alpha\sigma h'(x_1)h'(x_2)(\sigma h(x_1)-h(x_2))}{(\sigma h(x_1)+h(x_2))^3}=0$$
However, besides this approach, I fail to see how the equation could be obtained. Rearranging for $x_1$ to express player $1$'s BR function the conventional way seems no option here either, since the function $h$ is undefined, and the BR function would also become much messier that the one quoted.
*Some of the working out is in a technical appendix elsewhere, which I can email.
