Conformal conjugacy derivative for the chordal Loewner equation
= Conformal conjugacy derivative for the chordal Loewner equation
{title2=$\dot\phi_t(\xi_t)=-3\phi_t''(\xi_t)$}
For conjugated <Loewner chains>, $\phi_t=\widetilde g_t\circ\phi\circ g_t^{-1}$ and $\widetilde\xi_t=\phi_t(\xi_t)$. The image capacity speed is $2\phi_t'(\xi_t)^2$. Differentiating this identity gives $\dot\phi_t(z)=2\phi_t'(\xi_t)^2/(\phi_t(z)-\phi_t(\xi_t))-2\phi_t'(z)/(z-\xi_t)$. The removable singularity at the driver has value $-3\phi_t''(\xi_t)$.