I want to "work symbolically", but not only with a function, but also with derivatives. For example, I would like

f'[x] - f'[-x]

to simplify to zero.

Commentary: None of the ways suggested here helped me.

They seem to work, but now you need to create rules for Derivative not before f . And since Derivative it is not protected, you can do something like:

Derivative

f

Derivative[n_][f][-x_] := -Derivative[n][f][x] f''[x] + f'''''''[-x] + f''[-x] + f[x] + f'''''''[x]

f[x]

this is normal?