How to find the derivative of: \$s(t)=(1-e^{\frac{-t}{RC}})u(t)\$ w.r.t \$t\$?
Use Product Rule.

$$\frac{d}{dt}(1-e^{\frac{-t}{RC}})u(t)=\frac{d}{dt}(1-e^{\frac{-t}{RC}}).u(t)+\frac{d}{dt}u(t).(1-e^{\frac{-t}{RC}})$$
$$=(0+\frac{1}{RC}e^{\frac{-t}{RC}}).u(t)+\delta(t).(1-e^{\frac{-t}{RC}})$$
\$\delta(t)\$ is defined only at t = 0. And its value = \$\infty\$ at t = 0. Therefore ,
$$= \frac{1}{RC}e^{\frac{-t}{RC}}.u(t)+\infty (1-e^0)$$
$$= \frac{1}{RC}e^{\frac{-t}{RC}}.u(t)+\infty .0 $$
$$= \frac{1}{RC}e^{\frac{-t}{RC}}.u(t)$$