Hello,

Firstly I would like to say that I am new to this forum. I do not go to university yet, but I am sure this is university level mathematics. I have a quite specific question, and I am not sure it belongs to this sub-section of the forum. If it does not, then I apologize.

Calculation of Ellipse Arc Length

This website described the process of calculating the arc length of an ellipse. I understand everything up to the calculus part. Why exactly is the derivative of the arc length that specific value? It seems to somehow come from the Pythagoras theorem, but I do not quite see why the derivatives should be connected in such a way.

If you scroll down the webpage, a nice formula appears: (the ds/dx part)

Again, this seems to follow from the Pythagoras theorem, and again I do not know why. Why is dx/dx taken? Why are derivatives used at all?

And finally, once this is solved, how can I solve the integral to get the definite value of the arc length of part of the ellipse?

All credit from the website goes to the author of it.