Engine failure and rudder use
Like others have already posted: use the amount of rudder that you need to keep tracking in the direction that you desire!
Still it is possible to indicate what to expect:
As you are still rolling on the runway, with all wheels on the ground, you have the tracking force of nosewheels, main wheels and also the sideforce of the rudder available.
At rotation, you lose the tracking force of the nosewheels, so you will have to apply a bit more rudder.
As the main wheels leave the runway, their tracking force is also lost, so you will again have to apply a bit more rudder.
As you gain speed during your climbout, you will not need to adjust rudder force, because your rudder ratio changing sytem will adjust the actual deflection of the rudder for you (assuming that you talk about commercial size jetliners).
Of course, if you change thrust, either increase from reduced T/O thrust to FULL thrust, or decrease thrust from T/O to Max Continous, you will have to adjust the amount of pedal force.