Definition:Derivative/Vector-Valued Function/Point

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

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

Also see

 * Differentiation of Vector-Valued Function Componentwise: justification for this definition