# Unit step response and unit impulse response

In my ODE course I learnt that for a time invariant operator $$p(D)$$, if $$p(D)x=q(t)$$ then $$p(D)\dot{x}=\dot{q}(t)$$.

Then the professor "justified" that the derivative of the unit step response, v(t), is the unit impulse response, w(t), because the derivative of unit step function is the Dirac delta function.

However, w(t) and v(t) requires rest initial condition. There is no consideration of initial condition in the "justification" he gave.

Any thoughts?