Let $K$ be the rational function field $k(x)$ over a perfect field $k$ of characteristic $p > 0$. Let $F = k(u)$ for some $u\in K$, and write $u = f(x)/g(x)$ with $f$ and $g$ relatively prime. Show that $K/F$ is a separable extension if and only if $u\notin K^p$.
There is already a post on this but it has not yet been answered : Patrick Morandi- Field and Galois Theory- Section 4- Exercise 11
The truth is that I am very confused with this problem, I would like someone please help me to understand what happens with this exercise and explain to me how it could be done by giving me a help or something. I already have an approach for the left-to-right implication, but I do not know very well how good it is. Could someone help me with the other implication please? Thank you very much.