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