# Differentiation of Vector-Valued Function Componentwise

## Theorem

Let:

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

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

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

## Proof

 $\ds \frac {\d \map {\mathbf r} t} {\d t}$ $=$ $\ds \lim_{\Delta t \mathop \to 0} \frac {\map {\mathbf r} {t + \Delta t} - \map {\mathbf r} t} {\Delta t}$ Definition of Derivative of Vector-Valued Function $\ds$ $=$ $\ds \begin {bmatrix} \ds \lim_{\Delta t \mathop \to 0} \dfrac {\map {r_1} {t + \Delta t} - \map {r_1} t} {\Delta t} \\ \ds \lim_{\Delta t \mathop \to 0} \dfrac {\map {r_2} {t + \Delta t} - \map {r_2} t} {\Delta t} \\ \vdots \\ \ds \lim_{\Delta t \mathop \to 0} \dfrac {\map {r_n} {t + \Delta t} - \map {r_n} t} {\Delta t} \end {bmatrix}$ Definition of Limit of Vector-Valued Function $\ds$ $=$ $\ds \begin {bmatrix} \dfrac \d {\d t} \map {r_1} t \\ \dfrac \d {\d t} \map {r_2} t \\ \vdots \\ \dfrac \d {\d t} \map {r_n} t \end {bmatrix}$ Definition of Derivative of Real Function

$\blacksquare$