Automatic control - linearization with two states
I believe you are aware of the concept of the Jacobian matrix. This is a case of Linearization of Multistate Models. See, for example, pages 6-7 of 
Linearization of Nonlinear Models.

Yes, that linearization uses the calculation of the partial derivatives of  three functions with three variables (two states \$x_1\$, \$x_2\$ at one input \$u\$) around the equilibrium point \$Q\$. The definition of the Taylor series expansion (neglecting the h.o.t. - high-order terms) already suggests the use of the deviation variables:
$$\Delta x_1 = x_1-x_{1_Q}$$
$$\Delta x_2 = x_2-x_{2_Q}$$
$$\Delta u = u - u_Q$$

with which the final linearized model is built. Also, note that, for example:
\$\dot{\Delta x}=\dot{x_1}\$, since \$\dot{x_{1_Q}}=0\$.