$\begin{aligned} f(h) &\approx f(0) + h \cdot F(0) \\ f(2h) &\approx f(h) + h \cdot F(h) \\ f(3h) &\approx f(h) + h \cdot F(2h) \end{aligned}$
$\begin{aligned} f(h) &\approx f(0) + h \cdot F(\frac{h}{2}) \\ f(2h) &\approx f(h) + h \cdot F(\frac{3h}{2}) \\ f(3h) &\approx f(h) + h \cdot F(\frac{5h}{2}) \end{aligned}$
$\begin{aligned} f'(x) &= - \frac{1}{x^2} \\ x_{n+1} &= x_n - \frac{f(x_n)}{f'(x_n)} \\ &= x_n - \frac{\frac{1}{x_n} - b}{-\frac{1}{x_n^2}} \cdot \frac{-x_n^2}{-x_n^2} \\ &= x_n - (-x_n + bx_n^2) \\ &= 2x_n - bx_n^2 \\ &= x_n \cdot (2 - bx_n) \end{aligned}$