Day nine is called Navmi and is associated with Devi Siddhattrii or the goddess of salvation which is why even the deities worship Devi Siddhatrri. According to the puranas Lord Mahadev worshipped the goddess to claim more powers and she obliged him apparently his Ardhanareshwar avatar is a result of a boon from Goddess Siddhatrii. The puja followed on this day is called Maha Navmi Puja and is also recognized as Kanya puja popular all over India.
Dussera is the final day of the festival and celebrated differently in different states. In Gujrat the emphasis is on goddess Amba and Bahuchara, in Punjab it is Devi Shakti or Devi Sherawali. In Kerala it is Devi Saraswati and In Kashmir it is Devi Bhawani.
Dashami or Vijayadashmi is associated with the victory of Lord Rama when he vanquished Ravana in Ashok Vatika and left for Ayodhya after 14 years of exile. It is the happiest and most auspicious day in Hindu mythology and believed that anything started on the Dashami will become vijay/ successful for sure.