instead of approaching zero, then the right-hand side of the above equation would be written

Hence, the forward difference divided by approximates the derivative when is small. The error in this approximation can be derived from Taylor's theorem. Assuming that is twice differentiable, we have

The same formula holds for the backward difference:

However, the central (also called centered) difference yields a more accurate approximation. If is three times differentiable,

The

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