Derivative of Arc Length

Theorem
Let $C$ be a curve in the cartesian coordinate plane described by the equation $y = f \left({x}\right)$.

Let $s$ be the length along the arc of the curve from some reference point $P$.

Then the derivative of $s$ with respect to $x$ is given by:
 * $\displaystyle \frac{\mathrm d s}{\mathrm d x} = \sqrt{1 + \left({\frac{\mathrm d y}{\mathrm d x}}\right)^2}$

Proof 1
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.

Proof 2
From Continuously Differentiable Curve has Finite Arc Length, $s$ exists and is given by: