Electric Powertrain Torque-Speed Mapping (Electrical & Mechanical Engineering)

An electric vehicle (EV) powertrain engineer at UNLV is modeling the electromagnetic torque and efficiency characteristics of a permanent-magnet synchronous motor (PMSM) across its field-weakening operating regime. The motor mechanical output power $P(\omega)$ (in kilowatts) as a function of angular rotor velocity $\omega$ (in hundreds of radians per second, $\text{rad/s}$) is described by the multi-factor product function:

1. Compute the first derivative $\frac{dP}{d\omega}$ using Logarithmic Differentiation to bypass nested Product and Quotient rules.
2. Formulate the fractional power sensitivity function $\frac{1}{P}\frac{dP}{d\omega}$ and evaluate its exact numerical value at a cruising angular velocity of $\omega = 4.0\text{ (400 rad/s)}$.
3. Explain how powertrain engineers utilize the sign of $\frac{dP}{d\omega}$ and logarithmic sensitivity to optimize regenerative braking torque transition thresholds and prevent thermal battery inverter overload.