From what I remember in school and Wikipedia, the Newton-Raphson method was always very intuitive to me since:
$y=f'(x_n)·(x-x_n) + f(x_n)$ is basically saying take the slope for the small space in $x$ which will give you the corresponding change in $y$ and add that to the y that you currently have. Given some rearranging, we would get $$ x_{n+1} = x_n - f(x_n)/f'(x_n). $$
However, in reading about generalized linear modeling, the paper mentioned solving for large datasets using numerical techniques and the Newton-Raphson equation give was: $x_{n+1} = x_n - f'(x_n)/f''(x_n)$.
So I'm really asking for two things:
- What would be the intuitive explanation of taking the first derivative/second derivative? How should I interpret this?
- What's better about f'/f'' compared with f/f'?
Read again your new text. With almost certainty you will find that you want to find extremal points of that $f$. Since in general they can be found were $f'(x)=0$, Newton's method for $g(x)=f'(x)$ gives the iteration $$ x_{n+1}=x_n-\frac{g(x_n)}{g'(x_n)}=x_n-\frac{f'(x_n)}{f''(x_n)} $$