Boolean algebra derivative

Question

Assuming we have this function: $$F(x,y)=x^2+4y$$ we know that its pariale derivative is: $$d(x^2 + 4 y)/dx = 2 x$$ but if we have a mix of algebric function and boolean operators how can we calculate a partial derivative? ex: $$F(x,y)=x^2+4y+x⊕y$$ what I have found so far is that there is a branch of mathematics dealing with the concepts of differentials and derivatives of Boolean functions, however in this paper (https://www.anstuocmath.ro/mathematics/pdf17/RUDEANU-boolederiv.pdf) it turns out that boolean derivative is not the same as classical derivative. so how can we solve this issue?

Answer 1

It depends on how you've defined the operators $+, \cdot, \oplus,$ and $\odot$ so that you may combine them in this manner.

You need the partial derivative to work in the expected manner over $+, \cdot~$, for your specific $\oplus$ operation to work as the general one on page 179, and the partial derivative to be a "good derivative" .

If you've managed that, then identity (2) would claim:

$\qquad\dfrac{\partial [f(x,y)\oplus g(x,y)]}{\partial x} =\dfrac{\partial f(x,y)}{\partial x}\oplus g(x,y)+f(x,y)\oplus\dfrac{\partial g(x,y)}{\partial x}$

And in particular

$\qquad\dfrac{\partial [x\oplus y]}{\partial x} =1\oplus y+x\oplus0$