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Derive the following difference approximation for the first derivative:

$f'(x_0) = (f'(x_0 + 2h) - f(x_0 - h))/3h$

I really just need some pointers in how to start this out. If I were to guess, it looks like it starts out with the Lagrange form of the interpolating polynomial, differentiate f with respect to x, and when we get to

$f'(x_0) = (f'(x_1) - f(x_0))/(x_1 - x_0)$

We substitute $x_1 = x_0 + 2h$

But I'm not really sure if this is correct, and if it is, I'm not sure how to go about starting the derivation... If somebody could give me some pointers that'd be great. I'm also not sure which tags should be attached to this, so if anybody feels like editing it to add some better ones, that'd be great.

Thank you

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    Your difference quotient is really centered not at $x_0$, but rather at $x_0+h/2$ (the average of places where you evaluate $f(x)$ in getting the difference quotient. I guess if $f'(x)$ is assumed continuous that should be OK...2012-10-29

3 Answers 3