# Definition:Derivative/Vector-Valued Function

## Definition

### Derivative at a Point

Let $U \subset \R$ be an open set.

Let $\mathbf f \left({x}\right) = \displaystyle \sum_{k \mathop = 1}^n f_k \left({x}\right) \mathbf e_k: U \to \R^n$ be a vector-valued function.

Let $\mathbf f$ be differentiable at $u \in U$.

That is, let each $f_j$ be differentiable at $u \in U$.

The derivative of $\mathbf f$ with respect to $x$ at $u$ is defined as

$\dfrac {\d \mathbf f} {\d x} \left({u}\right) = \displaystyle \sum_{k \mathop = 1}^n \dfrac {\d f_k} {\d x} \left({u}\right) \mathbf e_k$

where $\dfrac {\d f_k} {\d x} \left({u}\right)$ is the derivative of $f_k$ with respect to $x$ at $u$.

### Derivative on an Open Set

Let $\mathbf r: t \mapsto \mathbf r \left({t}\right)$ be a vector-valued function defined for all $t$ on some real interval $\mathbb I$.

The derivative of $\mathbf r$ with respect to $t$ is defined as the limit:

 $\displaystyle \mathbf r' \left({t}\right)$ $:=$ $\displaystyle \lim_{\Delta t \mathop \to 0} \ \frac{\mathbf r \left({t + \Delta t}\right) - \mathbf r \left({t}\right)} {\Delta t}$ $\quad$ $\quad$

for all $t$ for which the limit exists.