Given the system $\dot{x} = f(x)$, $x(0) = x_0$, with $f \in C^{\infty}(\mathbb{R})$. I am trying to show that the flow map $\Phi(t, x_0)$ is invertible at any fixed point $x^{\ast}$ and at any finite time $t^{\ast}$. I was told that this would follow if I could show that $\Phi(t,x_0)$ is differentiable at $(t^{\ast}, x^{\ast})$ with respect to $x_0$ and $\frac{\partial \Phi(t^{\ast}, x^{\ast})}{\partial x_0} \neq 0$.
I am curious why differentiability of the flow map with respect to the initial condition at the fixed point guarantees invertibility and how one can show that in this case. My thought is that in order to use the smoothness of the vector field $f(x)$ I may want to expand $f(x)$ in a Taylor series about $x^{\ast}$.