This　paper　is　concerned　with　utilizing　higher-order　derivatives　of　generalized　Lyapunov-like　functions　to　infer　asymptotic　stability　of　the　switched　nonlinear　system　without　requiring　the　first　derivative　of　the　Lyapunov　function　for　each　activated　subsystem　to　be　negative　definite.　In　contrast　to　the　existing　switched　systems　literature,　we　allow　generalized　Lyapunov-like　functions　to　increase　both　during　the　continuous　flows　and　the　discrete　jumps.　Firstly,　a　set　of　sufficient　conditions　for　asymptotic　stability　of　switched　nonlinear　systems　are　derived　under　a　state-dependent　switching　law.　Then,　all　the　above-mentioned　conditions　are　transformed　to　a　set　of　bilinearly　diagonally　dominant　sum　of　squares　constraints.　Bilinearly　diagonally　dominant　sum　of　squares　programming　is　proposed　as　a　tool　to　solve　the　conditions　provided.　Finally,　two　examples　are　introduced　to　prove　the　effectiveness　of　the　above　method.