$\DeclareMathOperator{\tr}{tr}$ Are there any good references that have the identities of differentiation with respect to linear operators. More sepcifically, consider a finite-dim Hilbert space $\mathscr{H}$ over $\mathbb{R}$ or $\mathbb{C}$ (the 2 cases may be slightly different, I'm not sure yet) and let $\mathscr{L}(\mathscr{H})$ denote the linear operators on $\mathscr{H}$. Let $f:\mathscr{L}(\mathscr{H})\to \mathbb{C}$ be a "differentiable" function in the sense that there exists a linear operator $Df(A)$ such that $$ \frac{||f(A+\delta A)-f(A)- \tr(Df(A)\delta A)||}{||\delta A||} \to 0, \quad \delta A \to 0 $$
An example would be something like $f(A) = \langle\varphi |A \varphi \rangle$ so that $Df(A) = |\varphi \rangle \langle \varphi|$. This seems to be related to matrix calculus, and should be the natural generalization for basis-independent functions, but so far, I haven't found anything on this subject.