# Definition:Derivative/Vector-Valued Function/Open Set

Let $\mathbf r: t \mapsto \mathbf r \left({t}\right)$ be a vector-valued function defined for all $t$ on some real interval $\mathbb I$.
The derivative of $\mathbf r$ with respect to $t$ is defined as the limit:
 $\displaystyle \mathbf r' \left({t}\right)$ $:=$ $\displaystyle \lim_{\Delta t \mathop \to 0} \ \frac{\mathbf r \left({t + \Delta t}\right) - \mathbf r \left({t}\right)} {\Delta t}$
for all $t$ for which the limit exists.