For derivative controller transfer function, it's \$K_ds\$ but the Laplace Transform of \$\frac{df(x)}{dx}\$ is \$sF(s)-f(0)\$. For example if
$$f(x) = \frac{dg(x)}{dx}$$ hence its Laplace will be
$$F(s) = sG(s)-g(0)$$
If the transfer function of derivative is \$s\$ then
$$\frac{F(s)}{G(s)}=s \Leftrightarrow F(s) = sG(s)$$
how can inverse Laplace transform of \$F(s) = sG(s)\$ give \$f(x) = \frac{dg(x)}{dx}\$