Is differentiation with respect to a vector always defined componentwise?

Question

When one takes the derivative of a function $f$ along the direction of some vector $\mathbf{v}$, i.e. the directional derivative of $f$ along $\mathbf{v}$ this operation is defined componentwise, i.e. each component of $\mathbf{v}$ acts individually on $f$ to give a row vector, such that $$D_{\mathbf{v}}f=\nabla{f}\cdot\mathbf{v}$$ with $$\nabla{f}=\left(\frac{\partial f}{\partial x^{1}},\frac{\partial f}{\partial x^{2}},\frac{\partial f}{\partial x^{3}}\right)$$ Is this the case in general? That is, if one has a vector valued function $\mathbf{f}=\mathbf{f}(\mathbf{r})$, then is the derivative defined by $$\frac{d\mathbf{f}}{d\mathbf{r}}=\left(\begin{matrix}\frac{\partial f^{1}}{\partial x^{1}} &\cdots &\frac{\partial f^{n}}{\partial x^{1}}\\\frac{\partial f^{1}}{\partial x^{2}}&\cdots &\frac{\partial f^{n}}{\partial x^{2}}\\ \ldots &\ddots &\ldots\\ \frac{\partial f^{1}}{\partial x^{n}}&\cdots &\frac{\partial f^{n}}{\partial x^{n}}\end{matrix}\right)$$ Does this then extend in general?