In a charging R – C circuit, the electric current is given by i =...

Given the electric current $i = i_0 e^{-t/RC}$.

The rate of change of current is given by the derivative of $i$ with respect to time $t$:

$\dfrac{di}{dt} = \dfrac{d}{dt} \left( i_0 e^{-t/RC} \right)$

\dfrac{di}{dt} =i_0 e^{-t/RC} \left(-\dfrac{1}{RC} \right) = -\dfrac{i_0}{RC}$e^{-t/RC}$

To find the rate of change of current at $t = 0$, substitute $t = 0$ into the expression:

\left. \dfrac{di}{dt} \right|_{t=0} = -\dfrac{i_0}{RC}e^{0} = $-\dfrac{i_0}{RC}$