He was a famous scholar and the Vedanta teacher of Kashi. He was the disciple of Vasudeva Sarvabhauma of Nilachala. (Bhaktiratnakara 1.776-778)
The conclusions of Vedanta contained in Shrimad-Bhagavatam that were expounded by Shri Chaitanya Mahaprabhu to Sarvabhauma Bhattacharya in Puri had been in turn taught by the Bhattacharya to Madhusudana Vachaspati.
Under Shri Madhusudana Vachaspati, Shrila Jiva Goswami studied Vedanta, logic, philosophy and all kinds of scripture.
Taking leave of Nityananda Prabhu, Jiva went to the house of Tapana Mishra at Kashi.
In that sense Madhusudana was Jiva’s Vaidantika guru (Bhaktiratnakara). Thereafter Jiva went to Vrindavana and Rupa and Sanatana, Jiva’s uncles, happily received their nephew. Jiva then received diksha from Rupa Goswami.
Madhusudana Vachaspati is also considered guru of Shrila Sanatana Goswami, who is Rati Lavanga-manjari in Krishna Lila.