Correct notation for (partial) derivative evaluated in a given point Consider a function $f : \mathbb{R}^N \to \mathbb{R}.$
In general, we write the function in the from $f({\bf x})$, where ${\bf x} = [x_1, x_2, \ldots, x_N]^\top \in \mathbb{R}^N$.
Consider the partial derivative with respect to the $i$-th component of ${\bf x}.$ I would write:
$$\frac{\partial f}{\partial x_i} ~\text{or} ~ \frac{\partial f({\bf x})}{\partial x_i}.$$
What is the correct way to represent this derivative when evaluated in the point ${\bf y} \in \mathbb{R}^N$?
I don't feel so comfortable writing:
$$\frac{\partial f({\bf y})}{\partial x_i},$$
since the reader may miss the relationships between $x_i$ and the argument of the function.
Is
$$\left.\frac{\partial f({\bf x})}{\partial x_i}\right|_{{\bf x} = {\bf y}},$$
a more correct way to write the partial derivative evaluated in ${\bf y}$?
 A: If you want to be more precise, you can use the notation we use for the directional derivative and write $\frac{\partial f({\bf y})}{\partial e_i}$ where $e_i$ is the i-th versor of the canonical base of $\mathbb{R}^N$
in fact by definition you get that
$$\frac{\partial f({\bf y})}{\partial e_i}=\lim_{t\to 0}\frac{f(y+te_i)-f(y)}{t}$$
A: I for one am not fond of the notation $\frac{\partial f({\bf x})}{\partial x_i}$ in any context.  I much prefer either $\frac{\partial f}{\partial x_i}({\bf x})$ or $f_{x_{i}}({\bf x})$.  In each case it is clear that we are working with a partial derivative of $f$ evaluated at ${\bf x}$, and I think $\frac{\partial f({\bf x})}{\partial x_i}$ is ambiguous for the reason you've mentioned (whether evaluating at a particular $\bf y$ or not).  To answer your question, I think either $\frac{\partial f}{\partial x_i}({\bf y})$ or $f_{x_{i}}({\bf y})$ would suffice.  You could also write $\left.\frac{\partial f}{\partial x_i}({\bf x})\right|_{{\bf x} = {\bf y}},$ and you may want to in certain contexts, but I think in general that notation is a bit bloated.
