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.
Such a derivative is also known as an ordinary derivative.
This is to distinguish it from a partial derivative, which applies to functions of more than one independent variable.
In his initial investigations into differential calculus, Isaac Newton coined the term fluxion to mean derivative.