I hope that the following question is appropriate to ask here as it is not exactly research or original Mathematics but rather an enquiry for a reference or "standard method of proof":

Suppose we are given a smooth manifold $M$ (compact without boundary) and a family $g_\tau$ of Riemannian metrics on $M$ with corresponding Laplacians $\triangle_\tau$. We can then consider the trace of the heat operator, $$ \text{Tr}\,(e^{-t\triangle_\tau}) $$ and in particular one may want to take the derivative of the trace with respect to the parameter $\tau$. It turns out that one has $$ \frac{d}{d\tau}\text{Tr}\,(e^{-t\triangle_\tau})= \text{Tr}\,(-te^{-t\triangle_\tau}\frac{d\triangle_\tau}{d\tau}) $$ and whilst I can formally verify this computation (using basic manipulation rules of the exponential function) I would like to understand how to rigorously derive it. Hence if there is a justification for this a computation (or reference to read up on it), that would be very helpful. (For the reference it would be great if it be self - contained and not too advanced, perhaps at beginning graduate level). (Thanks!)



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