Given a nonlinear model: $\dot{x}=f(x)$

In practice, we need to both discretize and linearize the nonlinear model. The question is which procedure should be first?

(1) Method 1: Linearizing first, then discretizing

first linearizing: $\dot{x}=\frac{\partial f}{\partial x}x$

then discretizing: $x(t+1)=(I+\Delta t \frac{\partial f}{\partial x})x$

(2) Method 2: Discretizing first, then  linearizing

first discretizing: $x(t+1)=x(t)+\Delta t f[x(t)]$

then linearizing: $x(t+1)=(I+\Delta t \frac{\partial f}{\partial x})x$

Conclusions: the two methods are equivalent.