Prove the Second-Order Formula for the Fourth Derivative

[imsdal]

Hello there!

I'm having some issues with understanding how to prove this equation. Any tips or solution would really help me out!

. . .$\displaystyle f^{(iv)}(x)\, =\, \dfrac{f(x\, -\, 2h)\, -\, 4f(x\, -\, h)\, +\, 6f(x)\, -\, 4f(x\, +\, h)\, +\, f(x\, +\, 2h)}{h^4}\, +\, O(h^2)$

In advance, thanks for the help!

BR,
imsda

 

[Deleted member 4993]

Hello there!

I'm having some issues with understanding how to prove this equation. Any tips or solution would really help me out!

. . .$\displaystyle f^{(iv)}(x)\, =\, \dfrac{f(x\, -\, 2h)\, -\, 4f(x\, -\, h)\, +\, 6f(x)\, -\, 4f(x\, +\, h)\, +\, f(x\, +\, 2h)}{h^4}\, +\, O(h^2)$

In advance, thanks for the help!

BR,
imsda

Can you prove the equivalent form for the second derivative:

$\displaystyle \frac{d^2y}{dx^2} \ = \ \dfrac{y(x+h)-2y(x)+y(x-h)}{h^2}$