Definition:Derivative/Vector-Valued Function/Point

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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 $u$.

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