We study the non-linear conductance $mathcal{G}simpartial^2I/partial V^2|_{V=0}$ in coherent quasi-1D weakly disordered metallic wires. The analysis is based on the calculation of two fundamental correlators (correlations of conductances functional derivatives and correlations of injectivities), which are obtained explicitly by using diagrammatic techniques. In a coherent wire of length $L$, we obtain $mathcal{G}sim0.006,E_mathrm{Th}^{-1}$ (and $langlemathcal{G}rangle=0$), where $E_mathrm{Th}=D/L^2$ is the Thouless energy and $D$ the diffusion constant; the small dimensionless factor results from screening, i.e. cannot be obtained within a simple theory for non-interacting electrons. Electronic interactions are also responsible for an asymmetry under magnetic field reversal: the antisymmetric part of the non-linear conductance (at high magnetic field) being much smaller than the symmetric one, $mathcal{G}_asim0.001,(gE_mathrm{Th})^{-1}$, where $ggg1$ is the dimensionless (linear) conductance of the wire. Weakly coherent regimes are also studied: for $L_varphill L$, where $L_varphi$ is the phase coherence length, we get $mathcal{G}sim(L_varphi/L)^{7/2}E_mathrm{Th}^{-1}$, and $mathcal{G}_asim(L_varphi/L)^{11/2}(gE_mathrm{Th})^{-1}llmathcal{G}$ (at high magnetic field). When thermal fluctuations are important, $L_Tll L_varphill L$ where $L_T=sqrt{D/T}$, we obtain $mathcal{G}sim(L_T/L)(L_varphi/L)^{7/2}E_mathrm{Th}^{-1}$ (the result is dominated by the effect of screening) and $mathcal{G}_asim(L_T/L)^2(L_varphi/L)^{7/2}(gE_mathrm{Th})^{-1}$. All the precise dimensionless prefactors are obtained. Crossovers towards the zero magnetic field regime are also analysed.
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