with all one's might, as hard as one can; hard, diligently
一所懸命になって彼はその木に登った。
He climbed the tree with all his might.
もっと一所懸命勉強するって約束したよね。
You promised you'd study harder, remember?
私だって一所懸命なんだ
I'm trying my best too.
— 自転車泥棒 (1948)