proper notation for derivatives and differential This is a simple question (perhaps pedantic) about basic differentiation in real coordinate space.  In practice, I don't ever have issues using or carrying out differentiation in $\mathbb{R}^m$. But I'm pretty sure I frequently and severely abuse notation and I'm trying to improve.
question 1: Let $f:\mathbb{R}^m \to \mathbb{R}^n$ be some smooth map between real coordinate spaces. It maps some point $x\in\mathbb{R}^m$ to $y\;\dot{=}\;f(x)\in\mathbb{R}^n$.  What is the proper way to write the derivative of $f$ (Jacobian)?
$$
 df \quad, \quad \frac{\partial f}{\partial x} \quad, \quad \frac{\partial f(x)}{\partial x}  \quad, \quad
 \frac{\partial y}{\partial x} \qquad ?
$$
does $\frac{\partial f}{\partial x}$ even mean anything or do we need to "feed" $f$ some input before we can differentiate as $\frac{\partial f(x)}{\partial x}$? In the last of the above, $y$ is just a point, $y=f(x)\in\mathbb{R}^n$, not a function, that can be expressed in terms of $x$. Can we differentiate a point as $ \frac{\partial y}{\partial x}$ or is this simply a common abuse of notation?
question 2: continuing the above, what is the proper way to write the derivative of $f$ at a particular point, say $p\in\mathbb{R}^m$?  which of the following are correct, incorrect, or equivalent?
$$
 df(p) \quad, \quad \frac{\partial f}{\partial x}\big|_p \quad, \quad \frac{\partial f(x)}{\partial x}\big|_p   \quad, \quad \frac{\partial f(p)}{\partial p} 
\qquad?   
$$

I have a hunch that the "proper" notation for the derivative of some $f:\mathbb{R}^m \to \mathbb{R}^n$  is just $df$ and this is defined such that, at any arbitrary point $x\in\mathbb{R}^m$, it is given by $df(x)=\frac{\partial f(x)}{\partial x}$.  Is this correct?
context:  I posed this question in the context of real coordinate space(s), but I'm asking it with differential geometry in mind. My background is not in math and I recently started teaching myself some differential geometry and quickly realized I don't have a great grasp of proper mathematical notation.
edit: This question is related to one I asked on the physics page at this link
 A: $\frac{\partial f}{\partial x}$ always means a partial derivative, which means that $x$ is a specific coordinate with respect to some choice of coordinates of the domain. It's sloppy and confusing to use it to mean the total derivative and I would avoid doing that. I would write $\boxed{ df }$ for the total derivative and $\boxed{ df_p }$ for the total derivative at a point $p$. You can see this notation used e.g. on the Wikipedia article for total derivative.
I would avoid writing $df(p)$ because the total derivative is itself another function, namely a linear map, which takes as input a tangent vector $v$ at the point $p$; in my preferred notation this can be written $df_p(v)$ but if you want to use $df(p)$ you'd have to use the more awkward $df(p)(v)$; in my opinion this doubling of parentheses is hard to read and should be avoided.
I would also avoid calling the total derivative the Jacobian; they are conceptually not the same thing. The total derivative is a linear transformation and the Jacobian is a matrix describing that linear transformation with respect to a suitable choice of bases.
