What is the derivative of the function $s(t)=(1-e^{\frac{-t}{RC}})u(t)$?

Question

What is the derivative of the following function

$$s(t)=(1-e^{\frac{-t}{RC}})u(t)$$

with respect to $t$, where $u(t)$ is a unit step function?

I am getting $$\delta(t)+\frac{1}{RC}e^{-t/RC}u(t)$$ as the answer.

Is my answer correct?

Answer 1

I get \begin{align*} \dot{s}(t)&=\frac{d}{dt}\left[\left(1-\exp\left(-\frac{t}{RC}\right)\right)u(t)\right] \\ &=\frac{d}{dt}\,u(t)-\frac{d}{dt}\left[\exp\left(-\frac{t}{RC}\right)u(t)\right]\\ &=\delta(t)-\underbrace{\left[-\frac{1}{RC}\,\exp\left(-\frac{t}{RC}\right)u(t)+\exp\left(-\frac{t}{RC}\right)\delta(t)\right]}_{\text{Don't forget the product rule!}} \\ &=\left(1-\exp\left(-\frac{t}{RC}\right)\right)\delta(t)+\frac{1}{RC}\,\exp\left(-\frac{t}{RC}\right)u(t). \end{align*}