I need to compute the following derivative:
$$ \frac{\partial}{\partial V_{mn}} \int \prod_j d\phi_j \, \mathrm{Tr} \left(\ln(i \phi + \beta J V)\right) $$ where $\phi \equiv \phi_i \delta_{ij}$ is a diagonal matrix and $V$ is a matrix with elements $V_{mn}$ that are independent of $\phi_j$ (note the $i$ inside the natural log is the imaginary unit).
My attempt
1) I understand the $V_{mn}$ and $\phi_j$ are independent, so the derivative can be carried into the integral.
2) I'm less sure about whether I can carry the derivative into the trace -- but I assume so because of linearity.
Then I would think the answer would be (with chain rule):
$$ \int \prod_j d\phi_j \, \beta J (i\phi + \beta J V)^{-1}_{mn} $$
Is my reasoning/answer correct? Any hints appreciated.