Definition:Derivative/Vector-Valued Function/Open Set

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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:

\(\displaystyle \mathbf r' \left({t}\right)\) \(:=\) \(\displaystyle \lim_{\Delta t \mathop \to 0} \ \frac{\mathbf r \left({t + \Delta t}\right) - \mathbf r \left({t}\right)} {\Delta t}\) $\quad$ $\quad$

for all $t$ for which the limit exists.