Let \(k\) be a field and let \(f\in k[x]\) be nonzero.
\(f\) is separable if and only if \(\gcd(f, f') = 1\).
If \(f\) is irreducible, then \(f\) is separable if and only if \(f'\neq 0\).
If \(\ch k = 0\), every irreducible \(f\) is separable.
If \(\ch k = p>0\), an irreducible \(f\) is inseparable if and only if \(f(x) = g(x^p)\) for some \(g\in k[x]\).
A root \(\alpha\) of \(f\) in a splitting field is repeated if and only if \(f'(\alpha)=0\), so \(f\) has a repeated root if and only if \(f\) and \(f'\) have a common root, if and only if \(\gcd(f,f')\neq1\); the gcd does not change under field extension.
If \(f\) is irreducible, \(\gcd(f,f')\) is \(1\) or an associate of \(f\), and it is an associate of \(f\) if and only if \(f\divides f'\), which, since \(\deg f'<\deg f\), holds if and only if \(f'=0\).
If \(\ch k=0\) and \(\deg f\geq1\), then \(f'\) has degree \(\deg f-1\), so \(f'\neq0\).
Write \(f=\sum_i a_ix^i\). Then \(f'=\sum_i ia_ix^{i-1}=0\) if and only if \(p\divides i\) whenever \(a_i\neq0\), that is, if and only if \(f(x)=g(x^p)\) for some \(g\in k[x]\).
A reducible polynomial with nonzero derivative can be inseparable: \(f(x) = x^2(x-1)\in\QQ[x]\) has \(f'=3x^2-2x\neq 0\) and the repeated root \(0\).