RC Circuit Transient Surge (Electrical Engineering)

An electrical engineer is monitoring a discharging capacitor circuit in an automated industrial system. The voltage $V$ (in Volts) across the capacitor decays exponentially over time $t$ (in seconds) according to the transient decay function:

However, this specific circuit is linked to an industrial automated thermal controller where the system clock speed fluctuates based on ambient temperature shifts. The clock's time parameter varies dynamically according to the operating temperature $\theta$ (in degrees Celsius, $^\circ\text{C}$), modeled by:

1. Identify the composite rate relationship needed to calculate $\frac{dV}{d\theta}$ using the Chain Rule in Leibniz notation.
2. Differentiate the component functions $\frac{dV}{dt}$ and $\frac{dt}{d\theta}$.
3. Calculate the operational time $t$ when the temperature reaches $\theta = 2^\circ\text{C}$.
4. Determine the instantaneous rate of change of voltage with respect to temperature ($\frac{dV}{d\theta}$) at $\theta = 2^\circ\text{C}$.