The derivative path of a PID controller "predicts" the future error regarding the variable which needs to be controlled and reach a certain reference point. More specifically, the outcome of the derivative path is analogous to the rate of change of the error:
$$ error = reference - measurement $$
For example, if the error is quickly decreasing then the output of the derivative would produce a negative value which will then be added to the summation of the outputs of the proportional and integral paths resulting in a reduction of the total control signal. The reduction of the control signal generally means a reduction on the rate of change of the controlled variable (for example the velocity, if position needs to be controlled).
The gain $\ K_d $ corresponds to the weight the control engineer adjusts to this behavior regarding the derivative path. Increasing the derivative gain $\ K_d $ means that it is important for the system to react more aggressively to the rate of change of the error. Increasing $\ K_d $ means the output of the derivative path is larger which, as mentioned above, will reduce the total amount of the control signal, the value of rate of change of the controlled variable (velocity) will decrease and as a result the system will approach more slowly towards the desired reference point $\ \rightarrow $ $\ t_r $ increases (lower velocity) & $\ \%OS $ decreases (controller slows down the system before it reaches the reference point).
In conclusion, the derivative path of the PID controller determines how the system approaches the reference point, whether this happens fast or slowly, and correspondingly reduces or increases the total control signal following the below pattern:
- error decreases $\ \rightarrow $ output is negative
- error increases $\ \rightarrow $ output is positive
and the amplitude of the output depends on how fast the error changes.