I am trying to understand the following statement. We have operators $A_1 ,\ldots , A_k$ for $k\leq d$. Then suppose that
$$
\det (x_1A_1+\cdots +x_d A_d)=a_1x_1^d+\cdots +a_d x^d+\cdots a_{12\cdots d}x_1\cdots x_d
$$
We want to prove that
$$Str(D^p A_1\circ\cdots D^p A_d)=Tr((-1)^F D^p A_1\circ\cdots D^p A_d)=0 \quad \text{ if } k<d$$
and
$$ Str(D^p A_1\circ\cdots D^p A_d)=Tr((-1)^F D^p A_1\circ\cdots D^p A_d)=(-1)^{d}a_{12\cdots d} \quad \text{ if } k=d $$
The proof goes as following: \begin{align} det(I-e^{x_1A_1}\cdots e^{x_kA_k})=Str(\Lambda^p(e^{x_1A_1}\cdots e^{x_kA_k})) =Str(e^{x_1D^pA_1}\cdots e^{x_kD^pA_k}) \end{align} and the is says to compare the coefficients in both expessions (Str denotes the supertrace of an operator). Where $\Lambda$ and $D$ are $$ \Lambda^p A(u_1\wedge u^2\cdots \wedge u_p)=Au_i\wedge \cdots\wedge Au_p $$
$$ D^pA(u_1\wedge\cdots \wedge u_p)=\sum_{i=1}^{p}u_1\wedge\cdots \wedge u_{i-1}\wedge Au_i\wedge u_{i+1}\wedge\cdots\wedge u_p $$ This is from Patodi's proof of the McKean-Singer conjecture on his paper Curvature and eigenforms of the Laplacian. Any explanations or references on this would be greatly appreciated, especially on the first equation where the determinant becomes the Supertrace.