Rocket Drag and Aerodynamic Deceleration (Aerospace Engineering)

An aerospace propulsion engineer is modeling the aerodynamic drag force $F_D$ experienced by a sounding rocket ascending through the lower stratosphere. The aerodynamic drag (in kilonewtons) depends on the flight velocity $v$ (in $\text{m/s}$) and atmospheric density scale factors according to the empirical relationship:

for operational velocities $v \in [50, 1000]\text{ m/s}$.

1. Pre-process the radical denominator term $\frac{180}{\sqrt[3]{v}}$ into explicit power form $c \cdot v^n$.
2. Compute the derivative $\frac{dF_D}{dv}$, which represents the velocity sensitivity gradient of aerodynamic drag.
3. Factor out the lowest power from the resulting fractional terms to simplify the post-derivative expression, and evaluate $\left.\frac{dF_D}{dv}\right|_{v = 64\text{ m/s}}$.