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We explore thick branes in $f(R,\phi,X)$ gravity. We obtain the linear tensorperturbation equation of $f(R,\phi,X)$ branes and show that the branes arestable against the tensor perturbations under the condition of $\frac{\partialf(R,\phi,X)}{\partial R}>0$. In order to obtain thick brane solutions of thefourth-order field equations in this theory, we employ the reconstructiontechnique. We get exact solutions of the specific $f(R,\phi,X)$ thick branegenerated by a non-canonical scalar field. It is shown that the zero mode ofthe graviton for the thick brane is localized under certain conditions. Thisimplies that the four-dimensional Newtonian potential is recovered on thebrane. The effects of the Kaluza-Klein modes of the graviton for the$f(R,\phi,X)$ thick brane are also discussed.

Linear Stability of $f(R,\phi,X)$ Thick Branes: Tensor Perturbations