Negligent means indifference or careless about responsibilities or duties, and implies habitual behavior.
Essie is negligent about her studies.Negligible says that something is too insignificant or trifling to be concerned about.
After three years of hearing complaints about Fabian's negligence, I fired him.
Three dollars is too negligible an amount to concern me.
His paying attention to negligible details made it impossible for our team to complete our project.