Differentiation of Vector-Valued Function Componentwise

Theorem
Let:
 * $\mathbf r:t \mapsto \langle{f_1\left({t}\right),f_2\left({t}\right),\cdots,f_n\left({t}\right)}\rangle$

be a differentiable vector-valued function.

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


 * $\dfrac{\mathrm d\mathbf{r} }{\mathrm dt} = \langle{D_tf_1\left({t}\right),D_tf_2\left({t}\right),\cdots,D_tf_n\left({t}\right)}\rangle$