I will only add the note that the logarithmic derivatives studied by Shenghui in his post can also be expressed as
$$\frac{((n-1) x-n) x^n+x}{x (x-1) (x^n-1)}$$
Using the
$n=5$ case as an example,
With[{n = 5},
ComplexPlot[(((n - 1) x - n) x^n + x)/(x (x - 1) (x^n - 1)),
{x, 1.4}, ColorFunction -> "ShiftedCyclicLogAbs",
ImagePadding -> {{20, 3}, {15, 3}}]]
