Newton's Method

Newton's Method is a means of solving an equation for which there is no convenient closed form solution yet the first derivative of the equation is known. Assume the equation is of the form:
 * $y=f(x)$

And the value of x is wanted for a known y. Then an iterative improvement on an initial guess is of the form
 * $\displaystyle x_2=x_1-\frac{f(x_1)-y}{f^\prime(x_1)}$

Where $f^\prime(x)$ is the first derivative evaluated at $x$.

Proof
The function $f(x)$ can be expanded in Taylor's Expansion:
 * $\displaystyle f(x_2)=f(x_1)+f^\prime(x_1)(x_2-x_1)+\frac 1 2 f^{\prime\prime}(x_1)(x_2-x_1)^2+\cdots$

As $x_2$ gets closer to $x_1$, this series can be truncated to:
 * $\displaystyle f(x_2)=f(x_1)+f^\prime(x_1)(x_2-x_1)$

Let $x_\infty$ be the goal answer where:
 * $f(x_\infty)=y$

Let $\epsilon$ be the amount of difference from the new estimate and the goal answer:
 * $x_\infty = x_2 + \epsilon$

Then the function expanded around the new estimate is:
 * $y=f(x_2)+f^\prime(x_2)\epsilon$

Solving for $x_2$ produces:
 * $\displaystyle x_2=x_1-\frac{f(x_1)-y}{f^\prime(x_1)}-\epsilon\frac{f^\prime(x_2)}{f^\prime(x_1)}$