First law for the Kerr-Newman family ($dM=\kappa dA/(8\pi )+{\Omega }_{H}dJ+{\Phi }_{H}dQ$)

Differentiate $M=(r_{+}^2+a^2+Q^2)/(2r_{+})$, $J=Ma$ and $A=4\pi (r_{+}^2+a^2)$. Eliminating $d{r}_{+}$ and $da$ gives the displayed relation, with ${\Omega }_H=a/(r_{+}^2+a^2)$ and ${\Phi }_H=Q{r}_{+}/(r_{+}^2+a^2)$. Matching Hawking temperature to this First law of black-hole mechanics fixes the Bekenstein-Hawking entropy as $A/4$.