diligently
In a careful, steady way, with serious effort over time.
Diligently means working carefully, steadily, and with serious effort over time. When you work diligently, you stick with a task, pay attention to details, and keep trying until you get it right, even when it gets hard.
A student who studies diligently for a spelling bee doesn't cram the night before. She practices her words every day for weeks, testing herself, making lists of tricky spellings, and reviewing the ones she missed. A scientist works diligently in the lab, running the same experiment multiple times to make sure the results are accurate. When you practice piano diligently, you work on the difficult measures slowly, repeating them until your fingers know what to do.
Working diligently means caring about what you're doing. It involves focused, purposeful effort.
When teachers say they want you to work diligently, they mean they want to see you taking your work seriously, staying focused, and putting in consistent effort even when no one is watching. Diligent work often leads to real accomplishment, not because of talent or luck, but because of steady, careful attention over time.