# Definition:Derivative/Vector-Valued Function

## Definition

### Derivative at a Point

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

Let $\map {\mathbf f} x = \ds \sum_{k \mathop = 1}^n \map {f_k} x \mathbf e_k: U \to \R^n$ 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

- $\map {\dfrac {\d \mathbf f} {\d x} } u = \ds \sum_{k \mathop = 1}^n \map {\dfrac {\d f_k} {\d x} } u \mathbf e_k$

where $\map {\dfrac {\d f_k} {\d x} } u$ is the derivative of $f_k$ with respect to $x$ at $u$.

### Derivative on an Open Set

Let $\mathbf r: t \mapsto \map {\mathbf r} t$ 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:

\(\ds \map {\mathbf r'} t\) | \(:=\) | \(\ds \lim_{\Delta t \mathop \to 0} \frac {\map {\mathbf r} {t + \Delta t} - \map {\mathbf r} t} {\Delta t}\) |

for all $t$ for which the limit exists.

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## Also known as

Some sources refer to a **derivative** as a **differential coefficient**, and abbreviate it **D.C.**

Some sources call it a **derived function**.