I read through that and can find no mention of "x being held constant". In fact, because they take the derivative with respect to x, x can't be constant.
In the proof for Snell's law in physics, I don't understand why some variables are held constant and some not. Here's the proof:
Snell's Law - ProofWiki
They're taking the derivative of with respect to the length , as labeled in the picture above. Why in particular the length and not e.g. the length or perhaps ? How come all the other lengths are considered parameters i.e. why are we holding them in particular constant not in this context? Why do we choose and consider to be the actual variabel?
Note: I'm doing a similar assignment where I'm to maximize the area of a triangle and I'm having a bit of a hard time deciding with respect to which variabel I'm supposed to differentiate.
Hopefully I can be of some assistance. I understand the problem as follows:
1) A ray of light is emitted from A and is observed at B and there is no confusion about this. In other words, the points A and B are fixed: we know the light leaves A and is seen by B.
2) What we don't know is the path that the light takes from A to B. Luckily we have Fermat's Principle to help us with this!
3) What I think is confusing is that the artist of the diagram is not making it entirely clear what line represents the interface between the mediums - does he/she mean the solid line is the interface or the dashed line? I have concluded that they intend for the solid line to be the interface between the two because sits above the solid line and below it.
Since the solid line represents the interface, the "variable" a is constant because (using 1 above) A is fixed. Since A is fixed, its height above the solid line (represented by a) should not change. The "variable" b is constant for similar reasons.
From 2 above we recall that we don't know where the light wants to cross the interface. In other words we don't know how far to the right of A it crosses - the artist of the diagram has let this variable be represented by This is legitimate variable in the sense that we don't know what value for will minimize the time. Using Fermat's Principle we know that wherever it crosses will minimize the time. Everything else follows pretty much as stated in the link you posted.
So, I think the most important thing to realize is that since A and B are fixed, the variables and are constants, so we would not differentiate with respect to them.
Does this help answer your question? Let me know is anything is unclear.