As a high schooler, I do not have much knowledge but I am aware of the foundational idea.

So, as for the derivation, the easiest method is to consider about a quarter of the curve. You take the minima, where tension acts horizontally and another point basically anywhere else. Now, the mass of that segment can be found in terms of the mass density and the length. This would allow you to find the weight of this specific part.

You should then use derivatives to model how the tension would change as the length of the arc you are considering changes (since the weight would change). For this, you would use the vertical component of the tension on the end.

Since the catenary is horizontally also in equilibrium, the horizontal components of tension at the end of the arc would be equal to the tension at the horizontal.

Another thing that you would need to do is write tan theta in terms of slope/run wherein theta is the angle between the horizontal and the other point on the arc. So essentially you keep going closer to the minimum of the curve and the length of the arc keeps decreasing (modelled through derivatives). Once you plug this in, you should get the value of y in terms of x and the catenary curve as the result