I have a function $f(n,y)$ where $n$ is a nonnegative integer ($n\in \mathbb{N}_0$) and $y$ is a nonnegative real ($y\in \mathbb{R}_{\geqq 0}$). The following equation holds: $$ f(n,y) = a $$
where $a$ is a constant in $(0,1)$. I am interested in calculating the variations of $y$ with respect to $n$.
If the first variable were also continuous, i.e., $f(x,y)$ and $x, y, \in \mathbb{R}_{\geqq 0}$, I could apply implicit differentiation to the equation as follows: $$ \frac{\partial f}{\partial x} \frac{\partial x}{\partial x} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial x} = 0 $$
and then get the desired variations: $$ \frac{\partial y}{\partial x} = \frac{-\frac{\partial f}{\partial x}}{\frac{\partial f}{\partial y}} $$
But now since $n$ is discrete, I would replace the partial derivatives with the first forward difference with respect to $n$ in the above: $$ \Delta_n f + \frac{\partial f}{\partial y} \Delta_n y = 0 $$ $$ \Delta_n y = \frac{-\Delta_n f}{\frac{\partial f}{\partial y}} $$ where $\Delta_n f := f(n+1,y)-f(n,y)$.
However, I am not sure whether the above mixed-implicit difference-derivative calculation is correct. I have extensively searched books and the Internet, but most search results are about approximately solving differential equations and not explicitly on how the "mixed differentiation" should be defined and calculated. Any hints on how to approach this calculation using fundamental definitions and methods or references to relevant results are greatly appreciated.