Editing Secant method (section)

==Derivation of the method==
Starting with initial values {{math|''x''<sub>0</sub>}} and {{math|''x''<sub>1</sub>}}, we construct a line through the points {{math|(''x''<sub>0</sub>, ''f''(''x''<sub>0</sub>))}} and {{math|(''x''<sub>1</sub>, ''f''(''x''<sub>1</sub>))}}, as shown in the picture above. In point–point form,<ref>{{cite book |last1=Marsden |first1=Jerrold |title=Calculus I |date=1985 |publisher=Springer-Verlag New York Inc. |isbn=978-1-4612-5024-1 |pages=31 |url=https://link.springer.com/book/10.1007/978-1-4612-5024-1}}</ref> the equation of this line is

:<math>y = \frac{f(x_1) - f(x_0)}{x_1 - x_0}(x - x_1) + f(x_1).</math>
    
The root of this linear function, that is the value of {{mvar|x}} such that {{math|''y'' {{=}} 0}} is

:<math>x = x_1 - f(x_1) \frac{x_1 - x_0}{f(x_1) - f(x_0)}.</math>

We then use this new value of {{mvar|x}} as {{math|''x''<sub>2</sub>}} and repeat the process, using {{math|''x''<sub>1</sub>}} and {{math|''x''<sub>2</sub>}} instead of {{math|''x''<sub>0</sub>}} and {{math|''x''<sub>1</sub>}}. We continue this process, solving for {{math|''x''<sub>3</sub>}}, {{math|''x''<sub>4</sub>}}, etc., until we reach a sufficiently high level of precision (a sufficiently small difference between {{math|''x''<sub>''n''</sub>}} and {{math|''x''<sub>''n''−1</sub>}}):

:<math>
\begin{align}
x_2 & = x_1 - f(x_1) \frac{x_1 - x_0}{f(x_1) - f(x_0)}, \\[6pt]
x_3 & = x_2 - f(x_2) \frac{x_2 - x_1}{f(x_2) - f(x_1)}, \\[6pt]
& \,\,\,\vdots \\[6pt]
x_n & = x_{n-1} - f(x_{n-1}) \frac{x_{n-1} - x_{n-2}}{f(x_{n-1}) - f(x_{n-2})}.
\end{align}
</math>