D[g[x]*D[g[x]^(-1/2)*u[x], x], x]

Derivative[1][g][x]*
     (-((u[x]*Derivative[1][g][x])/
             (2*g[x]^(3/2))) +
        Derivative[1][u][x]/
          Sqrt[g[x]]) +
   g[x]*((3*u[x]*Derivative[1][g][x]^
               2)/(4*g[x]^(5/2)) -
        (Derivative[1][g][x]*
             Derivative[1][u][x])/
          g[x]^(3/2) -
        (u[x]*Derivative[2][g][x])/
          (2*g[x]^(3/2)) +
        Derivative[2][u][x]/Sqrt[g[x]])