We present the first experimental constraints on the spin-dependent WIMP-nucleon elastic cross sections from LUX data acquired in 2013. LUX is a dual-phase xenon time projection chamber operating at the Sanford Underground Research Facility (Lead, South Dakota), which is designed to observe the recoil signature of galactic WIMPs scattering from xenon nuclei. A profile likelihood ratio analysis of $1.4~times~10^{4}~text{kg}cdot~text{days}$ of fiducial exposure allows 90% CL upper limits to be set on the WIMP-neutron (WIMP-proton) cross section of $sigma_n~=~9.4~times~10^{-41}~text{cm}^2$ ($sigma_p~=~2.9~times~10^{-39}~text{cm}^2$) at 33 GeV/c$^2$. The spin-dependent WIMP-neutron limit is the most sensitive constraint to date.
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