 Weight-two transformation from an invariant Eisenstein limit

The finite Laurent coefficient $C(z)$ in the Kronecker limit formula is invariant under the modular group. Its Wirtinger derivative satisfies ${\partial }_{z}C=-\pi i(E_2(z)-3/(\pi y))/12$, by differentiating the convergent product in that formula. Differentiating $C(\gamma z)=C(z)$ gives weight-two covariance for $E_2^{\ast }$. Since $\Im (\gamma z)=y/|cz+d{|}^2$, substitution gives the displayed transformation law. This proof needs no prior modularity assertion for the eta product.