A derivative controller reacts to how quickly the error is changing:

$$u(t)=K_D\frac{de}{dt}.$$

In the ideal Laplace-domain model,

$$C(s)=K_Ds.$$

Derivative action can oppose rapid changes and add damping to a closed-loop response. It can reduce overshoot and improve transient behavior without directly accumulating steady-state correction.

Ideal differentiation also amplifies high-frequency measurement noise because its gain grows with frequency. Practical derivative action is therefore filtered, for example with a form such as

$$C_D(s)=K_D\frac{s}{1+s/\omega_f}.$$

Derivative control is best understood as a transient-shaping mechanism. It responds to the trend of the error rather than its accumulated history, and practical implementations must balance damping against noise sensitivity.