I read the example on the Wikipedia page on divided differences, but I don't understand why we do this "slope" thing $\frac{y_1-y_0}{x_1-x_0}$ between every known point, and why does it really solve our interpolation problem?
Because at this point I'm just plugging in the values without really understanding what's happening under the hood.
How do you explain this to yourself ? (maybe an image or any other intuitive approach)
we want a polynomial function of lowest degree that goes through $(x_0,y_0)$
$y = y_0$
Now we want it to go through $(x_1,y_1)$, too.
$y = y_0 + \phi_1(x-x_0)$
For any $\phi$ this line will still go through $(x_0,y_0)$ and we will need to find $\phi_1.$
$y = y_0 + \frac {(y_1-y_0)}{(x_1-x_0)}(x-x_0)$
Now we want it to go through $(x_2,y_2)$, too.
$y = y_0 + \frac {(y_1-y_0)}{(x_1-x_0)}(x-x_0) + \phi_2(x-x_0)(x-x_1)$ $y(x_2) = y_2 = y_0 + \frac {(y_1-y_0)}{(x_1-x_0)}(x_2-x_0) + \phi_2(x_2-x_0)(x_2-x_1)\\ \phi_2 = \frac {(y_2-y_0)}{(x_2-x_0)(x_2-x_1)} - \frac {(y_1-y_0)}{(x_1-x_0)(x_2-x_1)}\\ \phi_2 = \frac {\frac {(y_2-y_0)}{(x_2-x_0)} - \frac {(y_1-y_0)}{(x_1-x_0)}}{(x_2-x_1)}$
etc.