\begin{definition}[Definition:Derivative/Vector-Valued Function/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 $\mathbf f$ {{WRT|Differentiation}} $x$ 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 $u$.
\end{definition}