Definition:Derivative/Vector-Valued Function/Open Set

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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 $\mathbf r$ with respect to $t$ is defined as the limit:

\(\ds \map {\mathbf r'} t\) \(:=\) \(\ds \lim_{\Delta t \mathop \to 0} \frac {\map {\mathbf r} {t + \Delta t} - \map {\mathbf r} t} {\Delta t}\)

for all $t$ for which the limit exists.