I wish to give a slightly different answer compared to the others.
Strict and Standardized Notations is Very Important
They not only help us communicate better, they also help us think. They prime us to remember things and understand things better. For example, if I see $a^2 + b^2 = c^2$, I think Pythagoras Theorem and right angle triangles. If I see $k^2 + y^2 = t^2$, I don't.
It also allows you to be more accurate, and make sure your logic is not flawed.
Non-standard Notation Could also be helpful
Though rare, not using standard notations could help with thinking about a problem in another way, or coming up with a different sub-field of math.
There are Different Standards/Language is Evolving
Like any other language, mathematical language is evolving. If enough people uses a phrase, it is a correct phrase. Different mathematical papers uses different standards.
Commonly used Standard Notation could be suboptimal
The first notation for a subfield is usually made by the guy who ventures into this subfield. Being the first, he is exploring in unfamiliar territory, and his standards ends up suboptimal. Then more people come in, and each tries to invent a better standard, or a more universal notation, and it ends up like xkcd 927.
Furthermore, it should be noted that different notations are more useful in at different times.
Verbal Notations are often much more flexible than written ones
People often don't speak in completely correct sentences. Things are shortened. Words are changed.
Verbal Math is often an attempt to translate a formula to English?
How would you say $(2a + b) \times c$? There is no guide to speaking formula. Do you say "The product of c and the sum of two-a and b"? That clearly got the multiplicands in reverse, and what is a two-a? Or do you say "Open bracket, two times a plus b, close bracket, times c.
As another example, take "three x plus four b over seven all over nine". What does that mean?
So I'd say saying "d d x" or "d over d x" is perfectly fine. And if we can say that $f'$ is f prime; why can't be say that it is the prime of $f$, or that the action of differentiating is taking the prime of f?
Not Everyone Uses the Standard Notations/the same standard notations as you do
Unless your students will only be talking to you/other people that strictly follow the standards, they'll need to be flexible.
Are you sure you are right?
Are you sure that the things you find incorrect are actually incorrect, and not just using a particular standard?
Are you sure vertexes is not a allowable pluralization of vertex? Are you sure that the word vertices is not an appendage that is being/has been phased out? Will you insist that data must be plural, and one must use datum for the singular?
Are you sure you are pronouncing $\Omega$ correctly. Do you pronounce it like this or this? The former is more Greek, and is often used by people from certain areas in Europe (And, sometimes if taken to the extreme, sounds like "OH MY GOD"). Something like the latter is more used in America. The common pronounciation for me and my peers is something slightly different from the latter.
Are you sure the word "derive" cannot be used to mean "differentiate"? I cannot support this with evidence, but I remember some sources using derive in that manner, and some sources claiming that derive can indeed mean "differentiate".
You've already mentioned the debatability of inverse. I'm going to claim that using inverse of a fraction instead of reciprocal is perfectly allowed. And I would argue that "minus a from b" is perfectly allowed as well. "Plus a and b" is slightly more awkward. However, without consulting a mathematical grammar guide and dictionary, can you tell me why "plus" can not be used that way in math?
The Bottom Line
Everything considered, notations are important. You should seek to introduce your students to the different types of verbal notations. They definitely should be able to fluently use the word "differentiate". You should impress that some notations are more proper than others, and should be used most of the time. If called upon, they should be able to use proper notations.
However, it is also important for them to understand and use other "less proper" notations. In general, it is fine to use these "less proper" verbal notations. However, if it leads to a situation where the students are unable to use proper notation, or when the usage of certain verbal notation is hindering communication or thought, proper notation should be emphasized.
Finally, you can simply use the "correct" notation in your speech, and in general, the students will follow. You can also explicitly note, every so often, that while "derive" can be used sometimes, there are other notations, and "differentiate" is in a generally better and more clear word.
I feel the need to add onto this answer.
I would first like to draw attention to this question, which has great answers.
To quote some of the quotes given:
"The student of mathematics has to develop a tolerance for ambiguity.
Pedantry can be the enemy of insight." - Gila Hanna
As far as possible we have drawn attention in the text to abuse of
language, without which any mathematical text runs the risk of
pedantry not to say unreadability. - Bourbaki
Also linked in the answers to that question is an article by Terence Tao, who describes the progression of mathematical education in three stages: "pre-rigorous", "rigorous", and "post-rigorous". I'd argue that any sub-field in math is learnt kind of in this manner. I would say that the student should be only be steered toward correct notation in the pre-rigorous stage, and that if notation is to be emphasized, it should be during the "rigorous" stage.