Can somebody explain why you take the derivative of a curve to find its length?

Given a curve:

It goes between the two points:

To find its length:

What I understand is that you use the integral going from the values and to find the entire length between these two values. The squareroot and the exponential comes from the Pythagorean theorem to find the hypotenuse.

What I don't understand is why we take the derivative of the curve .

Re: Can somebody explain why you take the derivative of a curve to find its length?