Derivative of Arc Length/Proof 1

Proof
Consider a length $\mathrm d s$ of $C$, short enough for it to be approximated to a straight line segment:
 * DSbyDX.png

By Pythagoras's Theorem, we have:
 * $\mathrm d s^2 = \mathrm d x^2 + \mathrm d y^2$

Dividing by $\mathrm d x^2$ we have:

Hence the result, by taking the principal square root of both sides.