While reading Ordinary Differential Equations and Dynamical Systems by Gerald Teschl, I found the following statement:
"More Generally, suppose the equation $\dot{x} = f(x)$ in $\mathbb{R}$ has a fixed point $x_0$. Then it is not hard to see that $x_0$ is stable if \begin{eqnarray*} \dfrac{f(x) - f(x_0)}{x - x_0} \leq 0, \;\;\;\; x \in U(x_0)\setminus\{x_0\} \end{eqnarray*} for some neighborhood $U(x_0)$ and assymptotically stable if the inequality is strict."
This makes sense to me, as this expression is related to the derivative of $f$ (and consequently, the second derivative of the solution). As I understand, I am not supposed to calculate the eigenvalues of the Jacobian matrix, but I have to analyse the sign of $f$. Is there a way to show this expression this way?