do one's best
DO ONE'S UTMOST, try one's hardest, make every effort, do all one can, give one's all, bend over backwards, do one's damnedest, go all out, pull out all the stops, bust a gut, break one's neck, move heaven and earth

[TahlilGaran] Dictionary of Contemporary English ▲

do best
do (your) (level) best
to try very hard.
Tickets are hard to get, but I'll do my best to find you one.

Work is easy — you show up in the morning, do your level best for eight hours, then you go home.

do (your) level best
to try very hard to do something.
Tickets are quite hard to come by but I'll do my level best to get you one.
(often + to do something)
do (your) level best
to try very hard to do something.
Tickets are quite hard to come by but I'll do my level best to get you one.
(often + to do something)

[TahlilGaran] English Idioms Dictionary ▲

do one's best
to try to do something as well as one can
I tried to do my best on the exam.

[TahlilGaran] English Idioms Dictionary ▲

do one's best
v. phr. To perform at one's optimum capacity; spare no effort in fulfilling one's duties.
"I've really done my best teaching you people," the tired professor said on the last day of classes. "I hope you got something out of this course."

[TahlilGaran] English Idioms Dictionary ▲