This is the 11483rd most frequent Russian word.
должной
Proper, due, appropriate.
Here, 'должной' modifies 'ответственностью' to specify the type of responsibility, emphasizing it as due or necessary in the context.
Она подготовилась с должной ответственностью.
She prepared with due responsibility.
In this sentence, 'должное' acts as an adjective associated with 'внимание', highlighting the importance of providing the correct or appropriate level of attention.
Важно проявлять должное внимание к деталям.
It is important to give due attention to details.
'Должным' here qualifies 'уважением' to convey the idea of the respect being appropriate or required for the situation.