Differentiate an outer negative power

Let $u=6x^2+4x+1$ and $y=u^{-2}$. The source’s “outermost to innermost” guidance means differentiate the negative power first, then multiply by $u^{\prime}=12x+4$.

$dy/dx=-2(6x^2+4x+1)^{-3}(12x+4)=-8(3x+1)/(6x^2+4x+1)^3$. The denominator base is $6(x+\frac13)^2+\frac13>0$, so the expression and derivative exist for every real $x$.