Say I have a surface, x-y+xy+yz+z^2 = 0
(There is nothing particulary special about this expression. I've just made it too complicated to be simplified so that the variables are independent of one another)
Say I want to find all places on that function where x is constant. How do I find the equivalent to fx? (note that this is a surface, not a function. in other words it has a constraint that it always equals 0)
Let the left hand side of the equation defining your surface be denoted $F(x,y,z)$. The set of points where the plane tangent to the surface is parallel to the $y$-$z$ axis is the solution set of the triple of equations $$F(x,y,z)=0, \qquad \frac{\partial F}{\partial y}=0, \qquad \frac{\partial F}{\partial z}=0 $$
In your example, these three equations are $$x - y + xy + yz + z^2=0, \qquad -1+x+z=0, \qquad y + 2z = 0 $$ the solution set of which is easily found by substitution.