For small enough values of the function and for well-behaved functions, the terms beyond linear are unimportant, hence we obtain
$$ f(x)+(s-x)f'(x)\approx 0, $$yielding
$$ s\approx x-\frac{f(x)}{f'(x)}. $$Having in mind an iterative procedure, it is natural to start iterating with
$$ x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}. $$