DA40 Pilot reported airspeed indication was not working correctly on departure; resulting in a diversion to another airport.

Narrative

Take off on IFR flight plan from ZZZ to ZZZ1. Received Clearance and Release from ZZZ Departure Control. Conditions at the field where visibility poor/IMC. On takeoff roll airspeed came alive normally at around 40 kts. Upon descending sloped runway airspeed appear to slightly decline. Aircraft became airborne without control input and bounced at 42 kts. At this point we continued rotation and the airspeed indicator began to increase rapidly. Attempts to control airspeed with pitch resulted in activation of stall warning. Continued climb at known attitude and disregarded airspeed indication (both EFIS and backup show similar indications) to achieve known safe altitude.Found better visibility above fog layer at 3;000 ft. MSL. After evaluation [requested priority] and requested vectors to ZZZ2. Request ILS X do to IMC conditions. Airspeed indicator seems to function normally at level altitude but decreased rapid while descending; eventually ceasing function entirely and finally indicating failure per EFIS. Executed ILS based on known attitudes; descent rates; and power settings. Landing at ZZZ2 without further incident. Aircraft referred to local avionics shop for evaluation of the airspeed indicator and Pitot-Static systems. Apart for the initial malfunction no damage or injuries.