Let \( x^* \) be a fixed point, and let \( \eta(t) = x(t) - x^* \) be a small perturbation away from \( x^* \). To see whether the perturbation grows or decays, derive a differential equation for \( \eta \): \(\quad \boxed{ \dot{\eta} = \frac{d}{dt}(x - x^*) = \dot{x}}\)

\( \because \text{\( x^* \) is constant} \quad \therefore \dot{\eta} = \dot{x} = f(x) = f(x^* + \eta) \)
\( \therefore \text{Using Taylor’s expansion } f(x^* + \eta) = f(x^*) + \eta f'(x^*) + \mathcal{O}(\eta^2) \quad \because \text{\( x^* \) is a fixed point, } f(x^*) = 0 \)
\( \therefore \dot{\eta} = \eta f'(x^*) + \mathcal{O}(\eta^2) \)
\( \text{If \( f'(x^*) \neq 0 \), the \( \mathcal{O}(\eta^2) \) terms are negligible and we may write the approximation } \boxed{\dot{\eta} \approx \eta f'(x^*)} \)
\( \therefore \frac{d\eta}{\eta} = f'(x^*) \, dt \quad \therefore \int \frac{d\eta}{\eta} = \int f'(x^*) \, dt \Rightarrow \ln|\eta| = f'(x^*) t + C \)
\(\therefore |\eta| = e^{f'(x^*) t + C} = C_1 e^{f'(x^*) t} \Rightarrow \eta(t) = C_1 e^{f'(x^*) t}\)

The slope \( f'(x^*) \) at the fixed point determines its stability. The importance of the sign of \( f'(x^*) \) was clear from graphical approach; the new feature is that now we have a measure of how stable a fixed point is — that’s determined by the magnitude of \( f'(x^*) \). This magnitude plays the role of an exponential growth or decay rate. Its reciprocal \( 1/|f'(x^*)| \) is a characteristic time scale; it determines the time required for \( x(t) \) to vary significantly in the neighborhood of \( x^* \)