Definition:Derivative/Vector-Valued Function

At a Point
Let $U\subset\R$ be an open set.

Let $f = (f_1, \ldots, f_n)^\intercal : U \to \R^n$ be a vector-valued function.

Let $x\in U$.

Let $f$ be differentiable at $x$.

That is, let each $f_j$ be differentiable at $x$.

The derivative of $f$ at $x$ is $(f_1'(x), \ldots, f_n'(x))^\intercal$, where $f_j'(x)$ is the derivative of $f_j$ at $x$.

On an Open Set
Let $\mathbf r: t \mapsto \mathbf r \left({t}\right)$ 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:

for all $t$ for which the limit exists. .

Also see

 * Differentiation of Vector-Valued Function Componentwise