MHB Order of Accuracy for $\frac{f(x+2h)-f(x)}{2h}$ - Wave

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The divided difference $\frac{f(x+2h)-f(x)}{2h}$ approximates the derivative with an order of accuracy of 1. The derivation shows that after simplifying, the error term is proportional to $h f''(\xi)$, indicating the accuracy level. Participants in the discussion confirm the correctness of this conclusion. The consensus affirms that the method provides a first-order approximation of the derivative. This analysis is essential for understanding numerical differentiation techniques.
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Hello! (Wave)

With what order of accuracy does the divided difference $\frac{f(x+2h)-f(x)}{2h}$ approximate the derivative?

I have tried the following:

$$f(x+2h)=f(x)+ 2h f'(x)+ 2h^2 f''(\xi) , \xi \in (x,x+2h)$$

$$\frac{f(x+2h)-f(x)}{2h}=\frac{2h f'(x)+ 2h^2 f''(\xi)}{2h}=f'(x)+hf''(\xi)$$

$$\left|\frac{f(x+2h)-f(x)}{2h}-f'(x) \right|=h f''(\xi)$$

Thus the order of accuracy is $1$.

Am I right? (Thinking)
 
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