A nonlinear relation can be approximated by a linear one when the signal varies only slightly around a chosen operating point.

Suppose

$$y=f(x)$$

and the operating point is $(x_Q,y_Q)$. For a small perturbation $\Delta x$,

$$y\approx y_Q+f'(x_Q)\Delta x.$$

Writing only the varying part gives

$$\Delta y\approx f'(x_Q)\Delta x.$$

The derivative at the operating point is the local incremental gain. In electronic devices it becomes parameters such as transconductance or small-signal resistance.

A small-signal model replaces the nonlinear device by this local linear approximation while keeping the DC operating point conceptually separate.

The approximation is local. If the signal becomes large enough to move the device through substantially different parts of its nonlinear characteristic or into another operating region, the small-signal model no longer predicts the circuit accurately.