Differentiation of Vector-Valued Function Componentwise

Theorem
Let:
 * $\mathbf r: t \mapsto \tuple {\map {r_1} t, \map {r_2} t, \ldots, \map {r_n} t}$

be a differentiable vector-valued function.

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


 * $\dfrac {\d \map {\mathbf r} t} {\d t} = \tuple {D_t \map {r_1} t, D_t \map {r_2} t, \ldots, D_t \map {r_n} t}$