det(M) = det( ( A11 + B11, A12 + B12), (A21 + B21, A22 + B22) )

= det( ( A11 + 0, A12 + 0),  (A21 + B21, A22 + B22) )
+ det( ( 0 + B11, 0 + B12),  (A21 + B21, A22 + B22) )

= det( ( A11 + 0, A12 + 0),  (A21 + 0, A22 + 0) )
+ det( ( A11 + 0, A12 + 0),  (0 + B21, 0 + B22) )

+ det( ( 0 + B11, 0 + B12),  (A21 + 0, A22 + 0) )
+ det( ( 0 + B11, 0 + B12),  (0 + B21, 0 + B22) )

= det( ( A11, A12),  (A21, A22) )
+ det( ( A11, A12),  (B21, B22) )
+ det( ( B11, B12),  (A21, A22) )
+ det( ( B11, B12),  (B21, B22) )

det(M) =
= det( ( A11, A12),  (A21, A22) )
+ x det( ( A11, A12),  (C21, C22) )
+ x det( ( C11, C12),  (A21, A22) )
+ x^2 det( ( C11, C12),  (C21, C22) )

MatrixD[expr_, x__] := With[
    {old = OptionValue[SystemOptions[], "DifferentiationOptions"->"ExcludedFunctions"]},

    Internal`WithLocalSettings[
        SetSystemOptions["DifferentiationOptions"->"ExcludedFunctions"->Join[old, {Det, Inverse, Tr}]];

        Unprotect[D];
        (* handle list derivatives *)
        D[h:((Det|Tr|Inverse)[m_]), {z_, n_Integer}] := Nest[D[#, Replace[z, _List :> {z}]]&, h, n];
        D[h:((Det|Tr|Inverse)[m_]), {z_List}] := D[h, #]& /@ z;
        D[h:((Det|Tr|Inverse)[m_]), z_, y___] := D[D[h, z], y];

        (* define derivatives for Det, Tr, and Inverse *)
        D[Det[m_], z:Except[_List]] := Det[m] Tr[Inverse[m] . D[m,z]];
        D[Tr[m_], z:Except[_List]] := Tr[D[m,z]];
        D[Inverse[m_], z:Except[_List]] := -Inverse[m] . D[m, z] . Inverse[m],

        D[expr, x],

        SetSystemOptions["DifferentiationOptions"->"ExcludedFunctions"->old];
        Clear[D];
        Protect[D]
    ]
]

g[x_] := g0 + x g2 + x^2 g4 + x^3 g6 + x^3 hd Log[x];

coeffs = Table[
    MatrixD[Sqrt[Det[g[x]]] /. Log[x]->u, {x, n}]/n! /. {x->0, u->Log[x]},
    {n, 0, 3}
];
Print @* TeXForm /@ coeffs;

SeedRandom[2];
rules = Thread[{g0, g2, g4, g6, hd} -> RandomReal[1,{5,3,3}]]

N @ CoefficientList[Series[Sqrt[Det[g[x]]] /. rules, {x, 0, 3}], x]

coeffs /. rules //Expand