Step response of a differentiating system
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.