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)≈0,yielding
s≈x−f(x)f′(x).Having in mind an iterative procedure, it is natural to start iterating with
xn+1=xn−f(xn)f′(xn).