propagation of uncertainties of squared variables

Question

One rule of the propagation of uncertainties goes like this:
If $Q=A.B$ then $σ_Q=|A.B| √((σ_A/A)^2+(σ_B/B)^2 )$
so if $Q=A^2=A.A$ then $σ_Q=A^2.√(2.(σ_A/A)^2 )=|A.σ_A.√2|$.
However, another rule is: If $Q = A^b$ then $σ_Q=|b.A^(b-1).σ_A |$
so if $b = 2$ then $σ_Q=|A.σ_A.2|$. Which one is right?

Answer 1

If $Q=A.B$, you get what you said. But if $Q=A^2=A.A$, the answer is not simply the same as substituting $B=A$ (check here). What happens here is that, when B=A, then the 'independent variables hypothesis' is not valid anymore, and your answer is wrong.

Particularly, I prefer the general formula (applied to your problem) $$ \sigma_Q^2 = \left( \frac{\partial Q}{\partial A} \right)^2 \sigma_A^2$$ which gives as answer $ \sigma_Q = |2.A.\sigma_A|$