Given a curve:

$\displaystyle G(t)=\binom{t}{y(t)}$

It goes between the two points:

$\displaystyle A=(t_1,y_1) B=(t_2,y_2)$

To find its length:

$\displaystyle Length=\int_{t_1}^{t_2}\sqrt{\left(\frac{d}{dt}G(t )\right)^2}\, dt$

What I understand is that you use the integral going from the values $\displaystyle t_1$ and $\displaystyle t_2$ 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 $\displaystyle G(t)$.