A helicopter pilot reported an NMAC with a Cirrus that departed behind them and flew below them.

Narrative

I was the PIC and began the flight from ZZZ. The purpose of the flight was to conduct training with a helicopter-rated student. The aircraft was a Bell 429 helicopter. The student was at the controls. After startup; the student and I briefed a RWY XX departure. We were told by tower to initially hold after our takeoff request. Just over a minute later; we received our takeoff clearance; and it was to follow a departing Cherokee already rolling down RWY XX. The student began liftoff as the plane passed behind us. Our aircraft was facing the FBO with the tail toward the runway. We kept a visual on the Cherokee and the student began a hover taxi to the runway. The student; sitting in the right seat; verbally stated 'Clear right' before crossing the taxiway toward the runway and I verified visually as well. Then I looked to the left side of the aircraft on observed no traffic near us; on final or departing RWY XX. I announced 'Clear left' as we began to enter the runway adjacent to the FBO.As we entered the runway; we simultaneously began to increase speed and altitude. After climbing through approximately 150' to 200' AGL; the student began a small left turn toward the ZZZ VOR as to follow the Noise Abatement Procedure. At that same time the tower controller requested that we remain North of runway XX. Both the student and I had a moment of confusion. The student was already beginning to turn to the South. The student did reply to tower that we will stay to the 'North of the runway'. We very briefly questioned the tower's request between the two of us and I was about to give the instruction to turn the aircraft to the right and reposition to the North side of the runway when the student identified a departing Cirrus passing slightly below us and to the right. I quickly saw the Cirrus pass low and climb in front of us. I don't recall and further instructions to us from the tower; but the student did announce to me that the Cirrus was in sight. The Cirrus had made an early right crosswind turn and we were able to remain clear of that aircraft and the Cherokee ahead.We continued with a north turn inside the Cherokee. At this point we had not received any unusual instructions or communication from the tower; and I did not feel like it was the best time to contact them to discuss why the Cirrus overtook us on departure on the radio. I informed the student that I would contact the tower on the phone when we returned for an explanation and turned our discussion back to flying the aircraft. We returned to ZZZ Airport about 30 minutes later. After shutdown I called the tower supervisor. The supervisor stated that it appeared that we did nothing wrong. He also stated that there was training going on in the tower and that they probably didn't anticipate the slow departure of the helicopter. I believe there was a mistake in giving the Cirrus a takeoff clearance too early putting both aircraft near each other on the departure end of the runway.The tower's request for us to deviate from the airport's required noise abatement procedure caused confusion. I do not know at what point the Cirrus pilot had the helicopter in sight; but I do believe a collision was avoided due to the Cirrus seeing and avoiding the helicopter and keeping its flight path to the North of ours.