定义 设向量值函数$r = f\left( t \right)$在点${t_0}$的某一邻域内有定义，如果

$$\mathop {\lim }\limits_{\Delta t \to 0} \frac{{\Delta r}}{{\Delta t}} = \mathop {\lim }\limits_{\Delta t \to 0} \frac{{f\left( {{t_0} + \Delta t} \right) - f\left( {{t_0}} \right)}}{{\Delta t}}$$

存在，那么就称这个极限向量为向量值函数$r = f\left( t \right)$在点${t_0}$处的导数或导向量，记作

$$f'\left( {{t_0}} \right)$$

或

$${\left. {\frac{{{\text{d}}r}}{{{\text{d}}t}}} \right|_{t = {t_0}}}$$

向量值函数$f\left( t \right)$在${t_0}$可导（即存在导数）的充分必要条件是：$f\left( t \right)$的三个分量函数${f_1}\left( t \right),{f_2}\left( t \right),{f_3}\left( t \right)$都在${t_0}$可导；当$f\left( t \right)$在${t_0}$可导时，其导数

\[\begin{gathered} f'\left( {{t_0}} \right) = \hfill \\ {f_1}'\left( {{t_0}} \right)i + {f_2}'\left( {{t_0}} \right)j + {f_3}'\left( {{t_0}} \right)k \hfill \\ \end{gathered} \]

证明

$$\eqalign{ & \mathop {\lim }\limits_{\Delta t \to 0} \frac{{\Delta r}}{{\Delta t}} = \mathop {\lim }\limits_{\Delta t \to 0} \frac{{f\left( {{t_0} + \Delta t} \right) - f\left( {{t_0}} \right)}}{{\Delta t}} \cr & = \cr & \mathop {\lim }\limits_{\Delta t \to 0} \frac{{{f_1}\left( {{t_0} + \Delta t} \right)i + {f_2}\left( {{t_0} + \Delta t} \right)j}}{{\Delta t}} \cr & + \mathop {\lim }\limits_{\Delta t \to 0} \frac{{{f_3}\left( {{t_0} + \Delta t} \right)k}}{{\Delta t}} \cr & - \mathop {\lim }\limits_{\Delta t \to 0} \frac{{{f_1}\left( {{t_0}} \right)i + {f_2}\left( {{t_0}} \right)j + {f_3}\left( {{t_0}} \right)k}}{{\Delta t}} \cr & = \mathop {\lim }\limits_{\Delta t \to 0} \frac{{{f_1}\left( {{t_0} + \Delta t} \right) - {f_1}\left( {{t_0}} \right)}}{{\Delta t}}i + \cr & \mathop {\lim }\limits_{\Delta t \to 0} \frac{{{f_2}\left( {{t_0} + \Delta t} \right) - {f_2}\left( {{t_0}} \right)}}{{\Delta t}}j + \cr & \mathop {\lim }\limits_{\Delta t \to 0} \frac{{{f_3}\left( {{t_0} + \Delta t} \right) - {f_3}\left( {{t_0}} \right)}}{{\Delta t}}k \cr & = {f_1}'\left( {{t_0}} \right)i + {f_2}'\left( {{t_0}} \right)j + {f_3}'\left( {{t_0}} \right)k \cr} $$