If I want to use Newton's method for a polynomial function I simply use the definition and derive it to get to the fraction which needs to be deducted from the current guess. For instance, if $f(x)= x^3 - x - 1$, then I have $f'(x) = 3 x^2 - 1$.
Why can't I use $f(x)=\sqrt{x}$? I noticed that $\frac{f(x)}{f'(x)}$ gives me $-x$, which would not converge.
Thanks.
To overcome this inconvenience you can use a trick:
$$ f(x) := \lambda\sqrt{x} + (1-\lambda)(x-x_0) $$
where chosen a sufficiently large $n \in \mathbb{N}$:
$$ \lambda = \frac{k}{n}, \quad \quad \quad x_k = x_{k-1} - \frac{f(x_{k-1})}{f'(x_{k-1})} $$
with $k=1,2,\dots,n$.