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Formula Deduction
Linear coefficient at $$x_0$$:

$$a={f(x_0)-0 \over x_0-x_1}$$

Where: $$a=f'(x_0)$$

Thus $$f'(x_0)={f(x_0)-0 \over x_0-x_1}$$

$$x_0-x_1={f(x_0) \over f'(x_0)}$$

$$x_1-x_0=-{f(x_0) \over f'(x_0)}$$

Resulting in the formula for Newton's Method:

$$x_1=x_0-{f(x_0) \over f'(x_0)}$$