Derivative notation between $D$ and $\nabla$ Sorry if this is likely a duplicate question (and that I am asking two questions) but I am confused on some notation.
For a function $f(x_1, x_2): \mathbb{R}^2 \to \mathbb{R}$, for simplicity, why does the product rule tell us that
$$ \displaystyle Df(x_1, x_2) = \frac{\partial f(x_1, x_2)}{\partial x_1} \ dx_1 + \frac{\partial f(x_1, x_2)}{\partial x_2} \ dx_2 $$
Wouldn't we need a given functional form to know to apply the product rule (ie. if $f(x_1, x_2) = x_1 / x_2$ then we apply the quotient rule.
Further given the above, since the codomain of $f$ is scalar, it does not seem to match the following identity:
$$ \nabla f= [Df]^T $$
That is $Df$ is the transpose of the gradient of $f$, a vector.
(edit: I read these in Microeconomic Theory by Mas-Collel, Whinston and Green)
 A: That isn't the product rule. That is the exterior derivative of $f$. You know that $df$ is a 1-form with the property that if $v_p \in T_p\mathbb{R}^n$ then;
$$(df)(v_p) = v_p[f]$$
However since $df$ is a one form, if you select $x^1,...,x^n$ to be the standard coordinate functions on $\mathbb{R}^n$ then;
$$df = \sum_j \alpha^j \cdot dx^j$$
where $\alpha^j$ is smooth. Moreover, we know that if we let $p \in \mathbb{R}^n$, then with this selection of coordinates;
$$\left\{\frac{\partial}{\partial x^j}\Bigr|_p: j = 1,...,n\right\}$$
is a basis for $T_p\mathbb{R}^n$ with corresponding dual basis;
$$\{(dx^j)_p: j=1,...,n\} \Rightarrow dx^j\left(\frac{\partial}{\partial x^i}\Bigr|_p\right) = \delta_{ij}$$
Now if we apply $\dfrac{\partial}{\partial x^i}$ to both sides then we get;
$$(df)\left(\frac{\partial}{\partial x^i}\Bigr|_p\right) = \sum_j \alpha^j dx^j\left(\frac{\partial}{\partial x^i}\Bigr|_p\right) = \sum_j \alpha^j \delta_{ij} = \alpha^i$$
Looking at the LHS, by definition we have;
$$\left(\frac{\partial}{\partial x^i}\Bigr|_p\right)(f) = \alpha^i \Rightarrow df = \sum_j \frac{\partial f}{\partial x^j}\Bigr|_p \ dx^j$$
$\textbf{Addition}$: It may be just a choice of notation but maybe an expert can comment on the use for $\nabla$ notation. At least for a vector field $W$, the covariant-derivative of $W$ in the direction of a vector $\textbf{v}$ at $p$ is denoted as; $\nabla_{\textbf{v}} W = W(p+t\textbf{v})'(0)$. So in this context, $\nabla_{\textbf{v}}$ is a derivation(i.e tangent vector, operator) that gives you the initial-rate of change for a vector field in the direction $\textbf{v}$.
