I have an objective function $f(x_1,x_2,x_3)$ in an optimization problem, and don't know if all these variables are independent or not. Therefore, I want to know if $\dfrac{\partial x_2}{\partial x_3}$, or the partial derivative of $x_2$ with respect to $x_3$ is greater, less than, or equal to zero. Is total differentiation the right approach?
Or is $\dfrac{\partial x_2}{\partial x_3}$ equal to $\dfrac{\dfrac{\partial^2 f}{\partial x_3^2}}{\dfrac{\partial^2 f}{\partial x_2x_3}}$, and if so, why does this hold?