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:
 * $$\frac{\mathrm{d}{s}}{\mathrm{d}{x}} = \sqrt{1 + \left({\frac{\mathrm{d}{y}}{\mathrm{d}{x}}}\right)^2}$$

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 square root of both sides.