Linearization along "trajectory of fixed points" v.s. "arbitrary trajectory"

Question

I am reading a lecture and it says:

Consider the continuous nonlinear system: $$\dot{x}(t)=f(x(t)), \, x\in \mathbb{R}^n.$$

1. Linearize along arbitrary trajectory: $\bar{x}(t):[0,t] \rightarrow \mathbb{R}^n$. We get $\dot{\epsilon}(s)=A(s)\epsilon(s)$, where $$A(s)=D_xf(\bar{x}(s)).$$ Note that $f(\bar{x}(s))$ may not be $0$.
2. Linearize along trajectory of fixed points, i.e., $f(\bar{x}(s))=0$ for all $s\in[0,t].$ We get $$\dot{\epsilon}(s)=A\epsilon(s)$$

I am a bit confused about why $A$ does not depend on time $s$ for $s\in[0,t]$ in the second case?