RLC Circuit Dynamic Filter Transfer (Electrical Engineering)

An electrical communications engineer is designing a bandpass signal conditioning filter for a radio frequency telemetry receiver. The normalized output voltage gain $H(\omega)$ as a function of the angular frequency $\omega > 0$ (in $\text{krad/s}$) is described by the rational transfer function:

1. Apply the Quotient Rule using a formal Component Dictionary ($u, u', v, v'$) to find the frequency derivative $\frac{dH}{d\omega}$.
2. Simplify the numerator of the derivative by expanding and combining like terms.
3. Determine the cutoff frequency $\omega^*$ where the rate of gain change is zero ($\frac{dH}{d\omega} = 0$), corresponding to the resonance peak of the bandpass filter.