Let $R$ be a ring and $f(T) \in R[T]$ a monic polynomial over $R$. Then $f(T)$ is separable if $(f(T), \frac{d}{dt}f(T))=(1)=R[T]$ where $\frac{d}{dt}f(T)$ is the formal derivative of $f(T)$.
The example 3.4 from the Milne's book "Étale Cohomology" says:
$f$ is separable if and only if its image in $k(\mathfrak{p})[T]$ is separable for all prime ideals $\mathfrak{p}$ in $R$.
The "if" part is clear. Since there exist $\alpha , \beta \in R[T]$ such that $\alpha f +\beta \frac{d}{dt}f=1$ and its image equall to $1$ in $k(\mathfrak{p})[T]$, the images of $f$ and $\frac{d}{dt}f$ generates $k(\mathfrak{p})[T]$.
I do not understand the "only if" part.