GA instructor with student pilot reported an NMAC during takeoff from a XNX non-towered airport requiring evasive action to avoid a possible collision.

Narrative

I and my student landed on Runway 17 of XNX (Music City Executive Airport). After completing the landing we heard a pilot state they were lining up and waiting on Runway 17. We added full power and began a touch and go. As we were rotating we heard the other pilot say they were taking off. I could see on our ADS-B in that the airplane was already in a climb and gaining on our aircraft. We waited until we were at 1;300 MSL before making a crosswind turn. As we turned crosswind we did not see the aircraft. My student began to turn downwind and was unaware of any other aircraft. I took controls realizing the airplane that departed with us had turned early and chose to fly the crosswind in front of us. I took controls and started a hard right turn 360 degree on the left crosswind for 17. I made a radio call to tell the other aircraft they cut us off. They did not make any further radio calls.It was inappropriate for this pilot to take off while we were only a few feet off the runway. It was also inappropriate for them to turn left early knowing another airplane was in front of them. It was also inappropriate for them to fly as if they had right of way and cut in front of us from our left to right. We had the right of way as we were the traffic on the right.