In a manner of applying; with the characteristic of putting effort or dedication toward something.
From 'apply' plus the present participle suffix '-ing' plus the adverbial suffix '-ly,' creating an adverb describing the manner of applying oneself or something.
This is an extremely rare word—most English speakers would say 'diligently' or 'earnestly' instead—but it's technically grammatical and might appear in very formal or archaic writing.
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