# Numerical Recipes Eq. 9.4.6!

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1. The problem statement, all variables and given/known data

I want to derive equation Eq. 9.4.6 in Numerical Recipes from the expressions given, as stated in the book!
The equation represents the next (i+1 th) deviation $$\epsilon$$ from the true root.
Eq. 9.4.6:

$$\epsilon_{i+1} = -\epsilon_i^2 \frac{f''(x)}{2f'(x)}$$

2. Relevant equations

Eq. 9.4.5:

$$\epsilon_{i+1} = \epsilon_i + \frac{f(x_i)}{f'(x_i)}$$

$$\epsilon_i$$ represents deviation from true root.

General Taylor expansion:

Eq. 9.4.3:
$$f(x+\epsilon) = f(x) + \epsilon f'(x) + ...$$

$$f'(x+\epsilon) = f'(x) + \epsilon f''(x) + ...$$

3. The attempt at a solution

$$\epsilon_{i+1} = \epsilon_i^2 \frac{f''(x)}{f'(x) + \epsilon_i f''(x)}$$