inequalities – show this inequality with $frac{d^i}{dx^i}left(frac{x}{ln(1-x)}right)^{1/K} Bigg|_{x=0}>0, ~~~forall iin N^{+}$

Let $K$ be a fixed positive integer,show this $$dfrac{d^i}{dx^i}left(frac{x}{ln(1-x)}right)^{1/K} Bigg|_{x=0}>0, ~~~forall iin N^{+}$$

the problem is from when I solve this:$$left(sum_{i=1}^{n}a_{i}x^iright)^K=dfrac{x}{ln{(1-x)}},show ~that ~a_{i}>0,forall iin N^{+}$$.maybe this $$f(x)=dfrac{x}{ln{(1-x)}}$$ is special function? Is there a background to this conclusion?Thanks