  We derive the variational formula of the Loewner driving function of a simple
chord under infinitesimal quasiconformal deformations with Beltrami
coefficients supported away from the chord. As an application, we obtain the
first variation of the Loewner energy of a Jordan curve, defined as the
Dirichlet energy of the driving function of the curve. This result gives
another explanation of the identity between the Loewner energy and the
universal Liouville action introduced by Takhtajan and Teo, which has the same
variational formula. We also deduce the variation of the total mass of
SLE$_{8/3}$ loops touching the Jordan curve under quasiconformal deformations.
