Problem: Let $f: \mathbb{R} \to \mathbb{R}$ be an analytical function with a simple root $\alpha$. Given an iteration $$x_{r + 1} = x_r - \frac{f(x_r)^2}{f(x_r + f(x_r)) - f(x_r)},$$ prove that the order of convergence of a sequence $(x_n)_{n = 0} ^{\infty}$ is at least quadratic for $x_0$ sufficiently close to $\alpha$.
I've written $f$ in Taylor series around $\alpha$, since there's almost nothing else to do, but the convergence comes out linear. Any help would be greatly appreciated.