Residue formula for a quotient at a simple denominator zero (${\mathrm{Res}}_a(g/h)=g(a)/h^{\prime }(a)$)

If $g,h$ are holomorphic near $a$, $h(a)=0$ and $h^{\prime }(a)\ne 0$, write $h(z)=(z-a)k(z)$ with $k(a)=h^{\prime }(a)$. Then $g/h=(g/k)/(z-a)$, so its residue is $g(a)/h^{\prime }(a)$. If $g(a)\ne 0$ this is a simple pole; if $g(a)=0$ the singularity is removable and the formula gives zero.