Processor Thermal Power Dissipation (Computer & Electrical Engineering)

In modern computer microchip architecture, the local dynamic thermal power dissipation $P$ (in milliwatts, $\text{mW}$) of a processing core transistor sub-grid depends on its clock frequency $f$ (in gigahertz, $\text{GHz}$) and logic supply voltage $V$ (in volts, $\text{V}$), governed by:

To handle a demanding computational workload spike, an embedded Dynamic Voltage and Frequency Scaling (DVFS) microcontroller algorithm adjusts both operational parameters concurrently over run-time $t$ (in milliseconds, $\text{ms}$):

• $f(t) = 2 + \ln(t + 1)$
• $V(t) = 1.0 + 0.05t^2$

1. Compute the operating frequency $f(1)$ and voltage $V(1)$ at $t = 1\text{ ms}$.
2. State the multivariable chain rule formula for the total time derivative $\frac{dP}{dt}$.
3. Calculate the instantaneous rate of change of power dissipation $\frac{dP}{dt}$ at $t = 1\text{ ms}$.
4. Explain how modern thermal management firmware uses the partial sensitivities $\frac{\partial P}{\partial f}$ and $\frac{\partial P}{\partial V}$ to prevent processor thermal throttling.