\begin{definition}[Definition:Derivative/Vector-Valued Function/Open Set]
Let $\mathbf r: t \mapsto \map {\mathbf r} t$ be a vector-valued function defined for all $t$ on some real interval $\mathbb I$.
The '''derivative of $\mathbf r$ {{WRT|Differentiation}} $t$''' is defined as the limit:
{{begin-eqn}}
{{eqn | l = \map {\mathbf r'} t
      | o = :=
      | r = \lim_{\Delta t \mathop \to 0} \frac {\map {\mathbf r} {t + \Delta t} - \map {\mathbf r} t} {\Delta t}
}}
{{end-eqn}}
for all $t$ for which the limit exists.
\end{definition}