This question comes from S. Kobayashi's book ''Differential geometry of complex vector bundles''. Let $E$ be a complex vector bundle over a compact K$\ddot{a}$hler manifold. Fix an Hermitian structure $h$ in $E$. Let $D$ be a unitary connection in $E$. $R=D\circ D$ is the curvature and $K=i\Lambda R$, where $\Lambda$ is the trace operator. Let $\xi\in A^{1}(\rm{End(E,h)})$, $$ \partial_{t}R(D+t\xi)\big|_{t=0}=D\xi $$ then $$ \partial_{t}K(D+t\xi)\big|_{t=0}=D^{*}\xi. $$ So how can we obtain the above expression?
My computation is as follows:
$$ \partial_{t}K(D+t\xi)\big|_{t=0}=i\Lambda\partial_{t}R(D+t\xi)\big|_{t=0}=i\Lambda D\xi $$ Since $\Lambda D''\xi=-iD'^{*}\xi$ and $\Lambda D'\xi=iD''^{*}\xi$, $$ i\Lambda D\xi=i\Lambda (D'+D'')\xi=-D''^{*}\xi+D'^{*}\xi $$ But $D^{*}\xi=D''^{*}\xi+D'^{*}\xi$, where did I make a mistake? Thaks a lot.