Variation of operator valued functions
Consider some operator valued function \(F(\hat{A}) \), and its variation with respect to \( \hat{A} \):
$$
\delta F(\hat{A}) = F(\hat{A}+\delta \hat{A}) - F(\hat{A}).
$$
It is implicit that we work to linear order in \( \delta \hat{A} \). This quantity is nontrivial because arbitrary variations \( \delta \hat{A} \) in general won't commute with \( \hat{A} \). Supposing \( F \) is analytic, we can express it as a power series:
$$
F(\hat{A} + \delta \hat{A}) = \sum_{n=0}^{\infty} a_n (\hat{A} + \delta\hat{A})^n = \sum_{n=0}^{\infty} \big(\hat{A}^n + \hat{A}^{n-1}(\delta\hat{A}) + \hat{A}^{n-2}(\delta\hat{A})\hat{A} + \cdots + (\delta\hat{A})\hat{A}^{n-1}\big).
$$
We can read off the variation:
$$
\delta F(\hat{A}) = \sum_{n=0}^{\infty} (\hat{A}^{n-1}(\delta\hat{A}) + \hat{A}^{n-2}(\delta\hat{A})\hat{A} + \cdots + (\delta\hat{A})\hat{A}^{n-1}).
$$
This is a complicated expression, however life becomes much simpler if one considers instead the variation of the trace of \( F(\hat{A}) \):
$$
\mathrm{Tr}\big(\delta F(\hat{A})\big) = \mathrm{Tr}\bigg( \sum_{n=0}^{\infty} a_n n \hat{A}^{n-1} \delta \hat{A} \bigg) = \mathrm{Tr}\big(F'(\hat{A})\delta\hat{A}\big),
$$
where we used the cyclic property of the trace.