Superconformal transformation (Polchinski text 12.3.8)

I am reading Polchinski’s text book STING THEORY.
In the above of eq.(12.3.8), the differential $D_theta = partial_theta + thetapartial_z$ is defined and
begin{equation}
D_theta = D_thetatheta’partial_{theta’}
+ D_theta z’partial_{z’}
+ D_thetabartheta’partial_{bartheta’}
+ D_thetabar z’partial_{bar z’}.
end{equation}
I think it is given by the chain rule.
Next by using the property that superconformal transformation changes $D_theta$ into a multiple of itself, he derives $D_thetabartheta’ = D_theta z’ = 0$ and $D_theta z’ = theta’D_thetatheta’$. So $D_theta = (D_thetatheta’)D_{theta’}$.

Next he says $D_theta^2 = partial_z$ (I could prove it) and using it,
begin{equation}
partial_{bar z}z’ = partial_{bartheta}z’ = partial_{bar z}theta’ = partial_{bartheta}theta’ = 0
end{equation}
is obtained.

I cannot understand why this result can be derived.

P.S.
The solution of the above conditions is
begin{equation}
z'(z,theta) = f(z) + theta g(z)h(z),;;
theta'(z,theta) = g(z) + theta h(z),;;
h(z) = pm[partial_zf(z) + g(z)partial_zg(z)]^{1/2},
end{equation}
where $f(z)$ is ordinary holomorphic function and $g(z)$ is anticommuting holomorphic function.
He says "super-Jacobian", given in (A.2.29) as
begin{equation}
delta(dxdtheta)
= dxdthetaBig(sum_ifrac{delta}{delta x_i}delta x_i
+ frac{delta}{deltatheta_j}deltatheta_jBig),
end{equation}

of this transformation is
begin{equation}
dz’dtheta’ = dzdtheta D_thetatheta’.
end{equation}
How can I get it?