I have the following loci of points, and I need to find the scaling relationship that arises after I perturb $x_1$.
$\frac{\exp (x_1+x_2+x_3+x_4)}{T^2}\cdot\frac{(x_3-x_4)^2\left(e^{(x_1-x_2)/2T}+e^{-(x_1-x_2)/2T}\right)^2 - \frac{(x_1-x_2)^2}{2} (e ^{(x_3-x_4)/2T} +e^{-(x_3-x_4)/2T})^2 }{(e^{x_1/T}+e^{x_2/T})^2(e^{x_3/T}+e^{x_4/T})^2} - \delta =0$
where $\delta >0$. In other words, I have $$f(x_1,x_2,x_3,x_4)-\delta = 0$$
How do I perturb such a function to get a scaling relationship of for $x_1$? Most templates I see are of taking a Taylor expansion of some sort, but if I do simply $$f(x_1+\epsilon,x_2,x_3,x_4) -\delta = f(x_1,x_2,x_3,x_4) + \epsilon \frac{\partial f}{\partial x_1} - \delta \implies \epsilon\frac{\partial f}{\partial x_1} = 0$$
then the $\epsilon$ disappears and I do not get a scaling relationship.
What template can I follow to perturb a complicated locii of points to get the scaling relationship?
I would appreciate any advice you have for me!