Utilizing the approach recently proposed for the $\ell N$ inclusive scattering cross section on the lattice, we compute the differential scattering cross section for the charged current process $\ell p \rightarrow \nu n$ for various kinematical channels. The simulation is carried out on the $2+1$-flavor $16^3 \times 32$ ensemble with Iwasaki and Domain Wall Fermion action. The lattice results are compared with MINER$\nu$A result for the equivalent process.