Differentiation of Vector-Valued Function Componentwise

Theorem
Let:
 * $\mathbf r: t \mapsto \left\langle{r_1 \left({t}\right), r_2 \left({t}\right), \ldots, r_n \left({t}\right)}\right\rangle$

be a differentiable vector-valued function.

The derivative of a vector-valued function can be calculated by differentiating each of its component functions:


 * $\dfrac {\d \mathbf r \left({t}\right)} {\d t} = \left\langle{D_t r_1 \left({t}\right), D_t r_2 \left({t}\right), \ldots, D_t r_n \left({t}\right)}\right\rangle$