Conceptualising the continuous time unit impulse function as derivative of unit step
I didn't look at how the function $\delta_\Delta(t)$ is defined in that lecture, but the convention is that it's a rectangular pulse of area being $1$. The definition is :
$$\delta_\Delta(t) = \begin{cases}{ \frac{1}{\Delta} ~~~, ~~~ 0 &lt; t &lt; \Delta \\ ~0~ ~~~,~~~\text{otherwise} } \end{cases} $$
As you can see the area of that pulse, $A(\Delta) = \frac{1}{\Delta} \times \Delta = 1$ for any finite value of $\Delta$. According to calculus, that area is still $1$ even when the parameter $\Delta$ takes on inadmissible values of $0$ or $\infty$ through a limiting process. In particular the continuous Dirac delta function is defined as $\delta(t) = \lim_{\Delta \to 0} \delta_{\Delta}(t)$ ; when the limit of $\Delta$ goes to zero.
The rectangular pulse of unit area, $\delta_{\Delta}(t)$,  is only one of many other possible ways of defining $\delta(t)$ through a limit of functions process.