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## Homework Statement

I am asked to show that the formula:

f'(x)≈Ʃ[i=0,n] A

_{i}f(x

_{i})

which is derived from differentiating the interpolation polynomial

is similar to that derived from checking the order of accuracy formulae and similar to that derived through Taylor expansion.

## Homework Equations

## The Attempt at a Solution

I have found the error in all these cases to be of the form:

E(x)=C*f

^{(k)}*h

^{n}

which means that the error becomes zero for any polynomial of degree ≤ k-1.

Hence, for any such polynomial of degree ≤ k-1, the above formula could be rewritten thus:

f'(x)=Ʃ[i=0,n] A

_{i}f(x

_{i})

Now, as the error in the Taylor expansion would also be zero for such polynomial, f(x) could be accurately replaced with:

Ʃ[i=0,k-1] [f

^{(i)}(x

_{0})]*(x-x

_{0})

^{i}/i!

Thus, the formula would hold and be similar.

Is this attempt correct? Am I missing something? Would appreciate some advice.