We know the impulse response and the step response have the following relationship: $$h(t) = \frac{dy_{step}}{dt} $$
Given the following step response of a system:
$$ y_{step(t)} = \begin{cases} 0, \quad t\leq 0 \\ t, \quad 0\leq t \leq 1 \\ 1, \quad t\ge 1. \end{cases} $$
If I write out the step function using $u(t)$, then $y_{step(t)} = t*u(t) - t*u(t - 1)$.
Question 1: did I write out the function correctly?
Assuming it is, I proceed to take the derivative: $$h(t)=\frac{dy_{step(t)}}{dt} = \frac{d(t*u(t) - t*u(t - 1))}{dt}$$ $$ = u(t) + t\cdot\delta(t)-u(t-1)-t\cdot\delta(t-1)$$
Question 2: did I come up with the correct derivative?
Question 3: assume yes to Question 2, I know $t\cdot\delta(t)=0 \forall t$, but how do I deal with $t\cdot\delta(t-1)$?
p.s., the book answer show the $h(t) = u(t) - u(t-1)$
Thanks in advance.