Which of following sentence is the exact usage of the verb hate:
I hate my family not understanding me
I hate my family not to understand me
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The problem with both of your sentences is that, when you use the verb hate, the thing that you hate follows directly after the verb. Putting my family directly after the verb (even as part of the noun clause my family not understanding me) is confusing, because we parse it incorrectly as I hate my family then we are not sure what to do with the rest of the noun clause.
You can avoid this problem by putting something other than a noun directly after the verb. Here are some better ways of framing the same statement:
I hate that my family don't understand me. - with conjunction
I hate it that my family don't understand me. - with pronoun
I hate being misunderstood by my family. - with gerund
I hate to be misunderstood by my family. - with to+infinitive
Note that the last example does not sound particularly natural, but this format is definitely likely to be used with a would...
I would hate to be misunderstood by my family.
There are various ways to say this:
I hate (it) when my family doesn't understand me. (they frequently don't understand me)
I hate that my family doesn't understand me. (they usually don't understand me)
Both of these say that it's the situation you hate. If instead you want to say you hate the people (when they do something).
I hate my family when they don't understand me. (they frequently don't understand me)
I hate my family for not understanding me. (they usually don't understand me)
plus other expressions.
We can form these kind of "I hate X" (where X is some kind of noun clause) sentences as follows:
My family understands me
My family doesn't understand me.
I hate that (my family doesn't understand me.)
In the same way:
Jill is on time.
Jill is not on time.
I hate when (Jill is not on time.)
Note using "that", "when", "for" (or various others) can imply different things. For example, that suggests a recurring or permanent condition, while when implies an occasional or situational condition.