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i want to derive the following iterative scheme.

$$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)+f(x_n)} $$ where $n$ $=$ $0$$,$$1$$,$$.$$.$$.$

I think this scheme can be drive from Taylor's series but i didn't know how to approach. I already drive some iterative schemes using Taylor series like Householders,Halley's method and some other iterative schemes.How to approach to get this iterative scheme?

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What you have looks exactly like newton's method, except with an extra $f(x)$ in the denominator. One way to derive such a form would be to choose some $g(x)$ such that $(g f)' = g f' + g f$, which is, well, a clear sign of an exponential. So choosing $h(x) = \mathrm{e}^{x} f(x)$ and applying newton's method we have: $$x_{n+1} = x_n - \frac{h(x)}{h'(x)} = x_n - \frac{\mathrm{e}^{x} f(x)}{\mathrm{e}^x f'(x) + \mathrm{e}^x f(x)} = x_n - \frac{f(x)}{f'(x) + f(x)}$$

Note that since $\mathrm{e}^x$ "adds" no additional zeroes using newton's method in this way will still converge to $f(x)$'s zeroes.

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