I have a function in two variabels $f(n,x)$, one of the two variabels is discrete (i.e. $n\in\mathbb{N}$).
I want to solve the function/equation $f(n,x)=0$ wrt. $x$, i.e. $x=x(n)$ using the implicit function theorem and then find the dependence of $x$ on $n$. For a continuous variable this would be no problem with the implicit function theorem, i.e.
$$\frac{dx}{dn}=-\left(\frac{\partial f}{\partial n}\right)^{-1}\cdot \frac{\partial f}{\partial x}$$
Is there any discrete analogus to this theorem? E.g.
$$ x(n)-x(n-1) = -\frac{\frac{\partial f}{\partial x}}{f(n,x)-f(n-1,x)}$$
The proof for the existence of the function is not a problem I can show this independently from such a theorem (but would be nice to have of course)