Observation: Each click draws a vertical line to $(x_n, f(x_n))$, then slides down the Tangent Line back to the X-axis to find the next, better approximation $x_{n+1}$.

Warning: Try setting $x_0$ exactly at a local minimum (around $x=1.15$). The tangency becomes flat ($f'(x) \approx 0$), shooting your next guess off into infinity (Divergence)!