The Golden Ratio and the Secant Method The Secant Method is basically a way of replacing the second derivative of a function with an approximation. So $$f''(x) \approx \frac{f'(x_2)-f'(x_1)}{x_{2}-x_{1}}$$

We then use this in Newton's formula $$x_{k+1} = x_{k}-\frac{f'(x_k)}{f''(x_k)}$$

It turns out that the secant method converges with rate equal to the golden ratio. Is this just a coincidence?
It is not a coincidence.  It comes from the similarity from one stage of the root finding to the next and the fact that $\frac 1{\phi}=\phi-1$.  A good discussion is in section 9.2 of Numerical Recipes
The secant method replaces the first derivative, not the second.

Also, the Newton's method iteration is
$x_{n+1} = x_n - f(x_n)/f'(x_n)$.