I need to find the Laplace transform of $\dfrac{dh(t)}{dt}$ where $h(t) = e^{-100t} \cdot u(t)$ and $u(t)$ is the Heaviside function.
I have two ideas for solving this problem but I am unsure which is correct.
Idea 1: Take the derivative of $h(t)$ then take the Laplace transform of the result. So $dh(t)/dt = e^{-100t} \cdot \delta(t) - 100*e^{-100t} \cdot u(t)$
Idea 2: Use the laplace pair for a derivative as follows: $s\cdot \mathcal{L}\{h(t)\} - e^{-100t} \cdot u(0).$ The problem, however, is that $u(0)$ is undefined, so I'm not sure about how I would proceed with this method.
I would really appreciate some help in computing this Laplace transform.