What's the most fundamental derivative of multivariable functions? There are several derivatives of multivariable functions.  For instance, given a function $F: \Bbb R^n \to \Bbb R^m$ there's 


*

*the divergence $\nabla \cdot F$

*the curl $\nabla \times F$ (if $n=m=3$)

*the Jacobian $J_F(p)$

*the directional derivative $dF(p,v) = \lim\limits_{h\to 0}\frac{F(p+hv)-F(p)}{h}$


I'm wondering which of these is really the derivative of $F$.  I don't think it could be the divergence or curl because $(1)$ the curl isn't even defined except on $\Bbb R^3$ and $(2)$ when they're defined $(\nabla \cdot F)(p) = \operatorname{trace}(J_F(p))$ and $[(\nabla \times F)(p)]_k = \left[\frac12\big([J_F(p)]^T-J_F(p)\big)\right]_{ij}\varepsilon_{ijk}$.  So it seems that $J_F(p)$ is more fundamental.
But what about the directional derivative?  I don't even see have this relates to the Jacobian.  So which one (perhaps including some type of derivative I've never heard of) is more "fundamental"?  What is the derivative of a multivariable function?
 A: The Jacobian $J_F(p)$ contains all the data that all of the directional derivatives do. It tabulates the partial derivatives, which are directional derivatives in the directions $e_1, \dots, e_n$ (where $e_1, \dots, e_n$ is the standard basis of $\mathbb R^n$):
$$
J_F(p) = \begin{bmatrix}
\frac{\partial F_1}{\partial x_1}(p) & \cdots & \frac{\partial F_1}{\partial x_n}(p) \\
\vdots & \ddots & \vdots \\
\frac{\partial F_m}{\partial x_1}(p) & \cdots & \frac{\partial F_m}{\partial x_n}(p) \\
\end{bmatrix},
$$
where $F = (F_1, \dots, F_m)$,  $F_i : \mathbb R^n \to \mathbb R$ for $1 \leq i \leq m$. Any directional derivative $dF(p, v)$ (in your notation) is obtained from the partial derivatives by
$$
dF(p, v) = J_F(p) v.
$$
In particular, 
$$
dF(p, e_i) = J_F(p) e_i = \begin{bmatrix} \frac{\partial F_1}{\partial x_i}(p) \\ \vdots \\ \frac{\partial F_m}{\partial x_i}(p) \end{bmatrix}.
$$
As others have said, divergence and curl can be derived from partial derivatives.
However, if you asked me what the derivative of $F$ is, I would say none of these. I'd say the derivative of $F$ at $p$ is the linear map $A : \mathbb R^n \to \mathbb R^m$ such that
$$
0 = \lim_{h \to 0} \frac{F(p+h) - F(p) - A h}{\|h\|_\infty},
$$
if it exists, and I'd use the notation $A = DF(p)$.
The upshot here is that $DF(p)$ generalizes the nice properties of single-variable derivatives


*

*Differentiability $\implies$ continuity

*Composites of differentiable functions are differentiable,


while the directional derivatives (i.e., the Jacobian) do not. In fact, it can be the case that all directional derivatives exist at $p$ but $DF(p)$ as defined above does not exist. Moreover, if $DF(p)$ exists, its matrix is exactly $J_F(p)$. So I would call this the most general derivative of $F$.
This is an example (from Munkres's book Analysis on Manifolds, which I highly recommend) in which $F : \mathbb R^2 \to \mathbb R$ has directional derivatives at $0$ in all directions, but $F$ is not differentiable, or even continuous, at $0$:
$$
F(x,y) = \begin{cases}
\frac{x^2 y}{x^4 + y^2} & (x,y) \neq 0 \\
0 & (x,y) = 0
\end{cases}.
$$
A: The Jacobian $J_F(p)$ (using your notation) is just the matrix representation of the linear map $dF(p,\cdot)$. So they both are "the most fundamental" first derivative (in the sense that divergence, curl and similar operators can be derived from them).
A: Any function from $R^n$ to $R^m$, f(x), at $x= x_0$, can be written in the form $f(x)= A(x- x_0)+ \Delta$ where A is a linear transformation from $R^n$ to $R^m$ and $\frac{\Delta}{||x- x_0||}$ goes to 0 as $x$ goes to $x_0$.  Further, it can be shown that the linear transformation, A, is unique.  In the particular case that m= 1, A is a linear transformation from $R^n$ to $R$ which can be thought of as the dot product of an n dimensional vector with the n dimensional vector $x- x_0$.  That n-dimensional vector is $\nabla f$ or "grad f",  This precisely what florian said.
So the derivative, of a function from $R^n$ to $R$, at any point can be represented as an n dimensional vector which means that the derivative function is a function from $R^n$ to $R^n$ and so the second derivative at any point is linear function from $R^n$ to $R^n$ represented by an n by n matrix.  The third derivative at a point would be an "n by n by n" "matrix", etc.
