Proper use of various derivative and partial derivative notations

Question

I think I have seen all of these notations and more used

for derivatives:

$\dot f(x)$,

$f'(x)$,

$f_x(x)$,

$df(x)/dx$,

$D f(x)$

and for partial derivatives:

$\partial f(x,y)/\partial x$,

$\partial_x f(x,y)$,

$f_x(x,y)$,

$D_x f(x,y)$

and the corresponding notations used for higher order derivatives

for derivatives:

$f''(x)$,

$f_{xx}(x)$,

$d^2f(x)/dx^2$,

$D^2_x f(x)$

and for partial derivatives:

$\partial^2 f(x,y)/\partial x^2$,

$\partial_{xx} f(x,y)$,

$f_{xx}(x,y)$,

$D_{xx} f(x,y)$

I also think I have seen many of these used without the function argument, e.g $f'$ or $f_x$.

My question is: Is there an accepted proper usage for these derivative notations.

Any links or references would be appreciated.

See also Why are there so many notations for differentiation?

Answer 1

In Korner's A Companion to Analysis: A Second First and First Second Course in Analysis (pp. 396-397), he relates Felix Klein's complaint that

"There are a great many symbols used for each of the vector operations and, so far, it has proved impossible to produce a generally accepted notation. A commission was set up for this purpose at a scientific meeting in Kassel ($1903$). However, its members were not even able to come to a complete agreement among themselves. None the less, since their intentions were good, each member was willing to meet the others part way and the result was to bring three new notations into existence!"

This helped me realise that they are each used for a good reason, so you should make sure you are comfortable working between them as necessary. For the purposes of the reader though, it would be kind to stick to one notation at a time.