I am reading the following textbook:
Introduction to Applied Nonlinear Dynamical Systems and Chaos by Wiggins
p.6 (top)
Suppose we have $$\dot{x} = f(x,t), \ \ \ x\in \mathbb{R}^n $$
By implicit function theorem, if we can find $(\bar{x},\bar{t})$ such that $f(\bar{x},\bar{t})=0$ and $D_xf(\bar{x},\bar{t})\neq 0$, then we can find $\bar{x}(t)$, with $\bar{x}(\bar{t})=\bar{x}$ such that $f(\bar{x}(t),t)=0$.
If $\bar{x}(t)$ is a solution of the nonautonomous vector field, then it must be constant in time, i.e., $\dot{\bar{x}}(t)=0$.
Why this is the case? I am confused about it.