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 xn+1=xn−f(xn)f′(xn)[f′(xn)]2−f(xn)f″(xn). And then i need to show that f′(x)≠0. leading to F(x) has to be the same as f(x)? Then xn→r quadratically. A hint told me to use F'(r) does not equal to zero how do i use that?