Approximating Derivatives with Finite Differences

[Shade]

Homework Statement

Show that the first order derivative y'(xi) in the point xi may be approximated by

y'(xi)= (1/12*h) * (-3yi-1 -10yi + 18yi+1 -6yi+2 + yi+3) - (1/20h) h^4*y^(5) + O(h^5)

The Attempt at a Solution

I think the idea is to setup a linear system and some how use taylor expansion.

y'(xi) = a(-1)*y(xi-1) +
a(0) *y(xi) +
a( 1) *y(xi+1) +
a( 2) *y(xi+2) +
a( 3) *y(xi+3) +

Anyone has any idea on how I can show the above?

 

[HallsofIvy]

x_i is a point in how many dimensions? 2, 3?

 

[Shade]

I would assume one dimension.

xi are discrete points.

If anyone has any ideas on how to solve this please shout ;-)

 

[Shade]

If you can give a hint for n-dimensions HallsofIvy then I am sure I can solve it for 1d ;-)