I was trying to calculate the critical points of Yang-Mills functional. And I failed to show that $F_{A+ta}=F_{A}+t\nabla_Aa+t^2a\wedge a$. Here is my attempt: Suppose all calculation is in a local chart. Then, $$F_{A+ta}=d(A+ta)+(A+ta)\wedge(A+ta)=dA+A\wedge A+t(A\wedge a+a\wedge A)+t^2a\wedge a=F_A+t(da+A\wedge a+a\wedge A)+t^2a\wedge a.$$ However, I believe that it should hold $\nabla_Aa=(d+[A,-])a=da+A\wedge a-a\wedge A$, which takes the wrong sign.
Where is my fault? Could anyone help me please?