So it's easy to say that if $\alpha\in\mathbb{A}$ is a repeated root of $f(x)\in\mathbb{A}[x]$ then $\gcd(f',f)(\alpha)=0$. Conversely, also easy to verify that if $\beta$ is a simple root then it will no longer be a root of $f'$. For characteristic-p field, e.g. $\bar{\mathbb{F_q}}$, assuming the multiplicity not divisible by $p$ is sufficient for the equivalence to hold.
However, by degree counting, there might be some other $\gamma$ that is a root of $f'$ but not $f$. As long as the total number of original root decrease by more than 1, we will inevitably get new roots (under closure). Specifically, for characteristic-p field, the multiplicity is p multiple if and only if the $f'$ multiplicity will decrease. Otherwise, there might be additional multiplicities introduced through derivative of another component. For characteristic 0 field, the newly introduced roots is up to the number of irreducibles minus one, with multiplicity counted.
Getting here, one natural question to ask is that. Which roots exactly are being introduced by derivative? Is there an algebraically canonical way to describe them in terms of the original roots? Usually with topology added in, we could have geometric interpretation for this, but for general case it just not work.