to do your best, to try your hardest

一生懸命(いっしょうけんめい) Bust ass at xxx! adjective. 一生懸命頑張ってる (she’s really doing her best!).   $$$
 
努力(どりょく) Like 一生懸命 but a noun. one’s efforts.
 
必死(ひっし) Desperate, life-or-death, last-minute efforts.
 
奮闘(ふんとう) Stronger than 一生懸命 or 努力. Best defined as “epic struggle.” People struggle for justice in South Africa or for women’s right to vote. A struggle to keep one’s business going when a Wal-Mart moves in across the street.