Correct spelling, explanation: persistent is a common English adjective, but it originates in the Latin verb persistere, which used to mean to keep doing something with strength. The word entered English in the Middle Ages and today it is widely used in formal and informal contexts.

Definition of persistent:
adjective – lasting for a long time or is hard to get rid of,
I’ve been suffering from a persistent headache for weeks now.
adjective – determined,
Be persistent and never give up just because of laziness.

Persistent word family:
Other words that are from the word family of persistent include: persist (verb), persistence (noun), persistently (adverb)
If your headache persists, you need to see your doctor.
What brought her success was not only her talent but also her persistence in achieving goals.
She persistently ignores my remarks and still makes the same mistakes.

Incorrect spelling, explanation: two very common endings of English adjectives are -ent and -ant. As a result, many users of English get confused while using them. Exactly that happens in the case of persistent, which is often spelled as persistant. This form, however, is wrong so next time you write the word, make sure you spell it right.