Negation – Can “Not” Replace Words with Negative Prefixes?

Formally, it is the same prefix "in-".
However, there are four different spellings that help pronunciation.

The rule only depends on the first letter of the adjective:

L — use "il-": illegal, illiterate;
P, M, or B — use "im-": improper, immediate, imbalance;
R — use "ir-": irresponsible, irrational;
other consonants or vowels — use "in-": indifferent, innumerable, inadequate;

As @blubberguy says, "any" is not a negative word. Negative words are words like "no", "not", "never", and "none".

Like many "grammar rules" that you hear, "never use a double negative" is not really a valid rule. It is a guideline.

Yes, double negatives are sometimes just bad grammar. "I am not never gonna help Bob." What does that mean? Taken literally, it would seem to say that you ARE going to help Bob: "not" would mean to reverse "never".

But there are many cases where a double negative is perfectly valid grammatically. Sometimes we use a double negative for emphasis. "I am not, no way, ever going to help Bob." "Not" and "no" are both negatives, but they are clearly not reversing each other. They are repetition for emphasis.

Sometimes we want to reverse a negative. Like: "I am not refusing to help Bob." In this sentence, "not" and "refusing" are both negative words. But people often say things like this as a milder form of a positive. The person doesn't say he is "agreeing to help Bob", just that he is "not refusing". He may be helping Bob reluctantly, or he hasn't decided yet whether to agree or refuse. So he is not refusing, but he isn't necessarily actually agreeing either.

The problem with double negatives used this way is that it can be confusing. I recall a discussion of this that I read years ago where the writer mentioned a sentence he read in a newspaper about some political development that had a string of negatives. I forget the exact example but it was something like, "Congress decided not to refuse to delay the vote on repeal of the law banning X." This sentence has a number of words that in context are negative: "not", "refuse", "delay", "repeal", "banning". So was this action by Congress for or against X? You have to study it and think it through.

You often hear people say, "two negatives make a positive". Apparently they are trying to apply the rules of arithmetic to grammar, and are thinking of an expression like -(-3)=3. But who says the rules of arithmetic apply to grammar? And even if they do, sure, multiplying two negatives gives a positive, but adding two negatives gives a bigger negative.