Correct spelling, explanation: this popular English word comes from the verb persistere, which is Latin and meant to continue something despite difficulties. The word persistent appeared in Middle English and since then it has been widely used in both written and spoken language.

Definition of persistent:
adjective – something that lasts continuously,
I suffer persistent pain so I can’t handle a day without taking a painkiller.
adjective – stubborn and determined,
She is persistent and never gives up and that’s why she is so successful.

Collocations with persistent:
The word persistent usually collocates with the verb to be and adverbs, such as incredibly, very, and really.
He is incredibly persistent when it comes to running – he does it every day.
The headache is very persistent, it doesn’t go away even at night.
She is really persistent when it comes to her career so I expect her to get promoted soon.

Incorrect spelling, explanation: both learners and native users of English often spell the word persistent as persistant. It is the consequence of the fact that -ent and -ant are common adjective endings and as they are also similar, many users of English tend to make mistakes while spelling them. In the case of this particular word, however, the only correct form is persistent.