Atmospheric Density and Aerodynamic Drag (Aerospace Engineering)

During the vertical launch sequence of a research sounding rocket, the total aerodynamic drag force $F_d$ (in Newtons) acting against the vehicle casing depends directly on localized atmospheric air density $\rho$ (in $\text{kg/m}^3$) according to:

The atmospheric density decreases exponentially as a function of altitude $z$ (in kilometers) above sea level:

Furthermore, the rocket's vertical flight trajectory tracks altitude over burn-time $t$ (in seconds) according to the kinematic performance equation:

1. Deconstruct the three-layer composite relationship linking force $F_d$, density $\rho$, altitude $z$, and time $t$ using the Chain Rule in Leibniz notation.
2. Differentiate each structural layer independently to find $\frac{dF_d}{d\rho}$, $\frac{d\rho}{dz}$, and $\frac{dz}{dt}$.
3. Evaluate the rocket's altitude $z$ and vertical velocity $\frac{dz}{dt}$ at $t = 10\text{ seconds}$ into launch.
4. Calculate the dynamic rate of change of the structural aerodynamic drag force with respect to time ($\frac{dF_d}{dt}$) at $t = 10\text{ seconds}$.