I provide some context that involves game theory (but an in-depth understanding of this shouldn't be necessary for the mathematics, I think).
In some game, I have two players $1$ and $2$ playing contininious strategies $x_1$ and $x_2 \in \mathbb{R}_+$ respectively. In equilibrium, these strategies are optimal, in the sense that these maxmize their respective objective function (usually a profit function). The first order conditions (for maximization) for player $1$ and $2$ are given respectively by:
$$f(x_1,x_2,a,b)=1$$ $$g(x_1,x_2,a,b)=1$$
I'm interested in obtaining an expression for $\frac{\partial x_1}{\partial a}$, and I have come up with two different ways to go about obtaining this. I'm not sure which makes (more) sense?
Approach 1. Naively, we may apply the implicit function theorem:
$$\frac{\partial x_1}{\partial a}=-\frac{\frac{\partial f(x_1,x_2,a,b)}{\partial a}}{\frac{\partial f(x_1,x_2,a,b)}{\partial x_1}}$$
But given that $a$ is also a function of the first order condition for the second player, we must take into account the effect of $a$ on $x_1$ through its effect on $x_2$. Hence, I can refine this implicit differential as follows:
$$\frac{\partial x_1}{\partial a}=-\frac{\frac{\partial f(..)}{\partial a}+\frac{\partial x_2}{\partial a}*\frac{\partial f(..)}{\partial x_2}}{\frac{\partial f(..)}{\partial x_1}+\frac{\partial x_2}{\partial x_1}*\frac{\partial f(..)}{\partial x_2}}$$
Where $x_2$ satisfies player $2$'s FOC. In the numerator, the term $\frac{\partial x_2}{\partial a}*\frac{\partial f(..)}{\partial x_2}$ captures the effect of $a$ through $x_2$ on $f(..)$, and the $\frac{\partial x_2}{\partial x_1}*\frac{\partial f(..)}{\partial x_2}$ the effect of $x_1$ through $x_2$ on $f(..)$. Is this a correct application of the inverse function theorem that produces a good approximation of $\frac{\partial x_1}{\partial b}$? One factor that makes me hesitant about this PDE is in the numerator, the term $\frac{\partial x_2}{\partial a}*\frac{\partial f(..)}{\partial x_2}$ captures the effect of $a$ on $x_2$. However, this is exactly what we're trying to define all along, albeit for $x_1$.
Approach 2. Alternatively, we could differentiate both equations in $a$ to obtain
$$\frac{\partial f(..)}{\partial a}+\frac{\partial x_1}{\partial a}*\frac{\partial f(..)}{\partial x_1}+\frac{\partial x_2}{\partial a}*\frac{\partial f(..)}{\partial x_2}=0$$
$$\frac{\partial g(..)}{\partial a}+\frac{\partial x_1}{\partial a}*\frac{\partial g(..)}{\partial x_1}+\frac{\partial x_2}{\partial a}*\frac{\partial g(..)}{\partial x_2}=0$$
If we solve this system for $\frac{\partial x_1}{\partial a}$, we get
$$ x^1_a=\frac{f_2g_a-g_2f_a}{g_2f_1-g_1f_2}, $$
where I'm denoting $x^j_a=\partial x^j/\partial a$ and $f_j=\partial f/\partial x^j$.
I've been told that using using Jacobians, you can express this as $$ \frac{\partial x^1}{\partial a}=-\frac{\partial(f,g)/\partial(a,x^2)}{\partial (f,g)/\partial(x^1,x^2)}. $$
The nice thing about this method is that $x^2_a$ does not appear in this expression (in contrast to the first method)
Am I understanding/applying these methods correctly? Which method is better for expressing $\partial x_1/\partial a$?