Derive a formula for Newton’s method for the function $F(x) = f(x)/f′(x)$, where $f(x)$ is a function with simple zeros that is three times continuously differentiable. Show that the convergence of the resulting method to any zero r of $f(x)$ is at least quadratic.
So the answer I got for the first part is $x_{n+1} = x_n − \frac{f(x_n)f'(x_n)}{[f'(x_n)]^2 − f(x_n)f''(x_n)}$.
My thought process is that F(x) has the same root as f(x) so since F(x) is quadratic then f(x) is too? I dont know if im right. Thanks.