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こんなくそみたいな帽子{ぼうし}かぶせやがって！
"こんな仕事{しごと}させやがって！"
"こんなまずいものを食{く}わせやがって！"
"あいつは本当{ほんとう}に良{い}いものを作{つく}りやがる"
I was recently acquainted with the inflection "やがる," as illustrated in the examples above. Would it be right to describe it as a 活用形 (inflection)? Can anyone explain how to use it and what role it serves?
やがる is a verb since the endings it takes is the same as the other verbs:
食べやがる non-past
食べやがった past
食べやがれ imperative
etc.
やがる attaches to stems; it is an affix that attaches to a verb and creates a verb.
Its meaning is to add the first person's feeling that the act was done disfavorably or turned out to be disfavorable to the first person. It is the opposite of -てくれる, which expresses the first person's feeling that the act was favorable. Unlike -てくれる, though, やがる is colloquial and mildly insulting.
帽子を被せる
'put a hat on someone' [Neutral]帽子を被せてくれる
'put a hat on someone' [Favorable]帽子を被せやがる
'put a hat on someone' [Disfavorable]
In your last example, the reason やがる is used despite the positive connotation from よいもの is because the first person is expressing jealousy (serious or not). From the point of view of the first person, it is not a good thing that the other person made a good thing.
Traditional grammar confuses classification of words based on their meaning and classification based on the (morphological) form. It considers whatever word or morpheme that has the meaning comparable to the meanings expressed by auxiliaries in western languages (such as modaility, etc.) as 助動詞 'auxiliary verb'. However, parts of speech is a morphological (and syntactic) notion. It is a classification of words and morphemes based on their forms. It has nothing to do with the meaning. The traditional classification is inappropriate.
Furthermore, in modern analysis of Japanese, there is no such thing as 活用形. All there is is the verb stem, and an affix that attaches to it. What traditional grammar calls 活用語尾, on which 活用形 manifests, is actually the initial vowel of the affix (and/or some few sounds surrounding it). For example, traditional grammar says that the negative form verb 書かない consists of the stem 書, its 活用語尾 か and a 助動詞 ない. But notice how unsophisticated and complicated that analysis is. In modern analysis, all there is is the verb stem kak- and the negative affix -anai (which is itself an i-adjective).
やがる is a 助動詞 (auxiliary verb). 活用形 is used to refer to the conjugated form of a word.
やがる is used to state the action of the opponent with emotion of scorn or dislike.
From Yahoo dictionary:
軽蔑や憎しみなどの気持ちを込めて、相手の動作をいう意を表す
It is used after the 連用形 of verbs, and 「れる」「られる」「せる」「させる」.
