Hi,

my book starts to explain divided difference by saying that :

Suppose that is the nth Lagrange polynomial that agrees with the function at distinct numbers . The divided differences of with respect to are used to express in the form

Since I'm pretty slow, I do not see the logic behind this choice. I'm that I'm lacking some basic theory such that this appears like a mystical choice. Could someone point me in the right direction please?

Note:

I do understand why the interpolating polynomial between two points x_0 and x_1 can be written as,

The literature often states that a convenient form of the 2-order polynomial is,

How did man figure out that this is a convenient form? Trial and error? Thinking really hard?

Thanks.