Velocity

$${\bf v} \equiv {d{\bf r}\over dt},$$

where ${\bf r}$ is the Position Vector and $d/dt$ is the derivative with respect to time. Expressed in terms of the Arc Length,

$${\bf v} = {ds\over dt} \hat {\bf T},$$

where $\hat{\bf T}$ is the unit Tangent Vector, so the Speed (which is the magnitude of the velocity) is

$$v \equiv \vert{\bf v}\vert = {ds\over dt} = \vert{\bf r}'(t)\vert.$$

See also Angular Velocity, Position Vector, Speed