Differentiation of Vector-Valued Function Componentwise

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

be a differentiable vector-valued function.

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


 * $\dfrac{\mathrm d\mathbf{r}\left({t}\right) }{\mathrm dt} = \langle{D_tr_1\left({t}\right),D_tr_2\left({t}\right),\cdots,D_tr_n\left({t}\right)}\rangle$