Q:

Newton's method formula

I need to find the cube root of 9, using the Newton's method.  But firstly I need to write down the formula for $x_{n+1}$ which is $$x_n-\frac {f(x_n)}{f'(x_n)}$$, but I have to express $f′(x_n)$ in terms of $x_n$. But how to do it without the initial formula?

A:

If you're trying to find the formula for $x = \sqrt[3]{a}$, rewrite the function as $f(x) = x^3 - a$. Then, $f'(x) = 3x^2$. Thus, the Newton step formula is $$x_{n+1} = x_n - \frac{x_n^3 - a}{3x_n^2}$$ which is left as an exercise for you to simplify.

