In the context of statistical mechanics the "classical trace" is defined as $Tr(A e^{-\beta H}) = \int dr^N dp^N A e^{-\beta H}$ where $r^N$ and $p^N$ are phase space variables. So if $\Delta H$ is a small parameter and I wanted to linearized $Y = Tr(e^{-(H+\Delta H)}A)/Tr(e^{-(H+\Delta H)})$ I would expand $e^{-\Delta H}$ in a series order by $\Delta H$ and obtain $Y = Tr(e^{-H}(1-\Delta H + ...)A)/Tr(e^{-H}(1-\Delta H + ...))$. Here is where I am stuck. I know I need to multiply out the terms and collect all the terms to linear order but I do not know how to handle the fraction. Any help would be greatly appreciated as I am trying to fill in the blanks of the derivation of the Fluctuation-dissipation theorem given by Chandler.
The answer Chandler arrives at is $Y(t) = Tr(e^{-H}[A(t)-(\Delta H)A(t)+A(t)Tr(e^{-H}\Delta H)/Tr(e^{-H})])/Tr(e^{-H})+O(\Delta H)^2$