How to find x' and x'' using the finite difference approximation of the derivatives?

Question

I need to realize the modified Newton's method to find the minimum of my function. I have a block-schema of that method:

But my task requires to I used the finite difference approximation of the derivatives to find $f'(x)$ and $f''(x)$. I am trying to get how to find the derivatives from here and here. But my math level is awful really and the first is too hard for me. From the second article I got that: $$f'(x) \approx {f(x + h) - f(x) \over h}$$
Is it right? But what's $h$ in my case, $ε$?
How can I find $f''(x)$ from Taylor? $$f''(x) \approx 2*({f(x + ε) - f(x) \over ε^2} - {f'(x) \over ε})$$
Am I right? The problem is that it requires $f'(x)$ but if I will put an approximation of $f'(x)$ my $f''(x)$ will be zero.