Let $(E,h)$ be a hermitian holomorphic vector bundle and suppose we have local trivializations $U_i$ and $U_j$ with transition maps $g_{ij}$. Let $H_i$ and $H_j$ represent the matrices of $h$ with respect to the trivializations $U_i$ and $U_j$. We have that $H_j=g^T_{ij}H_i\bar{g}_{ij}$. I'm trying to prove the following equation
$$ \partial H_j\cdot H^{-1}_j=g^{-1}_{ij}(\partial H_i\cdot H_i)g_{ij} + g^{-1}_{ij}\partial g_{ij}, $$
but I've ran into some issues.
What I currently have is the following. We write $H_j=g^T_{ij}H_i\bar{g}_{ij}$ and then
$$ \begin{align*} \partial H_j=\partial(g^T_{ij}H_i\bar{g}_{ij})=(\partial g^T_{ij})H_i\bar{g}_{ij} + g^T_{ij}(\partial H_i)\bar{g}_{ij}+g^T_{ij}H_i(\partial \bar{g}_{ij}). \end{align*} $$
Now if I'm not mistaken $\partial \bar{g}_{ij} = 0$ so this reduces to
$$ \begin{align*} \partial(g^T_{ij}H_i\bar{g}_{ij})=(\partial g^T_{ij})H_i\bar{g}_{ij} + g^T_{ij}(\partial H_i)\bar{g}_{ij}. \end{align*} $$
Multiplying this thing from the right by the matrix
$$ H_j = (g^T_{ij}H_i\bar{g}_{ij})^{-1} = \bar{g}^{-1}_{ij}H^{-1}_i(g^T_{ij})^{-1} $$
we obtain
$$ \partial H_j\cdot H^{-1}_j=(\partial g^T_{ij})(g^T_{ij})^{-1} +g^{T}_{ij}(\partial H_i)H^{-1}_i\bar{g}_{ij}. $$
This is not quite what I wanted, but I can't figure out where am I making a mistake?
If $H_i = g_{ij}H_jg_{ij}^*$, then, writing $g=g_{ij}$ for ease of notation, we calculate as follows. Note that, as you said, $\partial g^* = (\partial\bar g)^\top = 0$, since $\bar\partial g = 0$. \begin{align*} \partial H_i\cdot H_i^{-1} &= \partial (gH_jg^*)(gH_jg^*)^{-1} = \partial g(H_jg^*(g^*){}^{-1}H_j^{-1})g^{-1} + g(\partial H_j)(g^*(g^*){}^{-1})H_j^{-1}g^{-1} \\ &= dg\cdot g^{-1} + g(\partial H_j\cdot H_j^{-1})g^{-1}, \end{align*} as desired.