Derivation of 2nd divided difference

[roldy]

I'm trying to understand how the second divided difference is formulated. I understand that the first divided difference is just the equation of a slope.

$f(x_{i},x_{i+1})=\frac{f(x_{i})-f(x_{i+1})}{x_{i}-x_{i+1}}$

Every source that I have read always jumps to the second divided difference by saying "and by induction"

$f(x_{i},x_{i+1},x_{i+2})=\frac{f(x_{i},x_{i+1})-f(x_{i+1},x_{i+2})}{x_{i+2}-x_{i+1}}$

How is induction used to get this equation?

 

[pari777]

http://nptel.iitm.ac.in/courses/Webcourse-contents/IIT-KANPUR/mathematics-2/node112.html

you can check the above site if you think it would be useful.

 

[HallsofIvy]

Having got two consecutive first differences,
$$\frac{f(x_i)- f(x_{i+1}}{x_i- x_{i+1}}$$
$$\frac{f(x_{i+1}- f(x_{i+2}}{x_{i+1}- x_{i+2}}$$

Now, for the first difference of those:
$$\frac{\frac{f(x_i)- f(x_{i+1}}{x_i- x_{i+1}}- \frac{f(x_{i+1}- f(x_{i+2}}{x_{i+1}- x_{i+2}}}{x_i- x_{i+2}}$$

 

[roldy]

Thank you. I new it was some type of substitution but I failed to think of using the first difference with the first differences.