If $u_s(t)$ is the unit step function, $\delta(t)$ the dirac delta function, and $f(t) = u_s(t)$, then $\frac {d} {dx}f(t) = \delta(t)$.
The laplace transform of $\frac {d} {dx}f(t)$ is $s(F(t))-f(0)$, meaning that $\frac {d} {dx}u_s(t) = s(\frac 1 s) - 1$, which is $0$.
The laplace transform of $\delta(t)$ is $1$.
Why do these values not match?