Small-amplitude AFM frequency shift ($\Delta \omega /{\omega }_0\simeq -F^{\prime }(z_0)/(2c)$)

At equilibrium separation $z_0$, expanding the tip force gives $m_{\mathrm{eff}}\ddot{x}=-[c-F^{\prime }(z_0)]x$. Thus $\omega /{\omega }_0=\sqrt{1-F^{\prime }(z_0)/c}$, and Taylor expansion gives the displayed result for $|F^{\prime }|\ll c$. The displacement convention takes positive $z$ away from the surface. A positive gradient of an attractive force softens the resonance; a large-amplitude oscillation samples more than the local gradient.