Every etale morphism is locally (passing to affine neighbourhoods and then to their coordinate rings) of the form $A \to (A[x]/(P(x)))_b$ where $P(x)$ has the property that $P'(x)$ is invertible in $(A[x]/(P(x)))_b$.
Is anything like that true for a general finite morphism? finite flat morphism?
(I am trying to understand the proof for the statement for etale morphisms from the stacks project, Algebra 133.16 and I cannot undersnand if the etaleness is crucial to the proof -- of course, one then doesn't need the condition on $P'(x)$ )