Inductor Transient Voltage (Electrical Engineering)

An electrical engineer tests a power regulation circuit containing a specialized high-frequency inductor with an inductance of $L = 0.5\text{ Henries}$ ($0.5\text{ H}$). During a rapid switching transient, the time-varying current $i(t)$ (in Amperes) flowing through the inductor for $t > 0$ seconds follows the rational decay function:

The fundamental physical constitutive relation governing the induced voltage $V_L(t)$ across an inductor is given by Faraday's law:

1. Using the limit definition of the difference quotient, compute the instantaneous rate of change of current $\frac{di}{dt}$ for any time $t > 0$.
2. Determine the algebraic formula for the induced inductor voltage $V_L(t)$ as a function of time.
3. Calculate the instantaneous voltage across the inductor at $t = 2\text{ seconds}$ and explain the physical significance of the negative sign.