Step response of a differentiating system

Question

Consider I can find the step response of a system with the following method:

We can easily find the step input of a system from its transfer function. Given a system with input $x(t)$, output $y(t)$ and transfer function $H(s)$

$$H(s) = \frac{Y(s)}{X(s)}$$

the output with zero initial conditions (i.e., the zero state output) is simply given by

$$Y(s) = H(s)X(s)$$

so the unit step response, $Y_\gamma(s)$, is given by

$$ Y_\gamma(s) = \frac{1}{s} H(s) $$

If my input is a "differentiating system" then what will be the step response ?

Answer 1

If your system is an ideal differentiator with input-output relation

$$y(t)=\frac{dx(t)}{dt}\tag{1}$$

then its transfer function is

$$H(s)=\frac{Y(s)}{X(s)}=s\tag{2}$$

From the equation in your question you obtain for its step response

$$A(s)=1\tag{3}$$

which in the time domain corresponds to a Dirac delta impulse:

$$a(t)=\delta(t)\tag{4}$$

This is also intuitively clear, because a step input has a derivative of zero everywhere except at $t=0$, where it is discontinuous.