One reason why the iteration
$$x_{n+1}=x_n-\tan\;x_n$$
converges quickly for appropriate starting values is that this is nothing more than the Newton-Raphson iteration for $\sin\;x$.
This got me thinking: given some arbitrary function $g(x)$, is there always a function $f(x)$ such that
$$\frac{f(x)}{f^\prime(x)}=g(x)$$
or are there restrictions on the nature of $g(x)$ so that the differential equation has a solution?