Prompt $J/\psi$ and $b\to J/\psi X$ production in $pp$ collisions

The LHCb measurement of the $J/\psi$ production cross-section is presented. The double differential cross-section is measured as a function of the $J/\psi$ transverse momentum $p_{\rm T}$ and of the $J/\psi$ rapidity $y$ in the fiducial region $p_{\rm T} \in [0;14],\rm {GeV}/c$ and $y \in [2.0;4.5]$. The analysis is based on a sample of 5.2 pb$^{-1}$ collected in September 2010 at the $pp$ Large Hadron Collider at CERN, at a centre-of-mass energy of $\sqrt{s}$ = 7TeV. The contributions from prompt $J/\psi$ and $J/\psi$ from $b$ are separated using the $J/\psi$ pseudo-proper time.