do one's best
Also, do one's level best or one's damnedest. Perform as well as one can, do the utmost possible, as in I'm doing my best to balance this statement, or She did her level best to pass the course, or He did his damnedest to get done in time. The first term dates from the 16th century, but the addition of level, here meaning "very," occurred only in the mid-1800s; the variant dates from the late 1800s.
Dictionary.com presents 366 FAQs, incorporating some of the frequently asked questions from the past with newer queries.