In one of the books I found that given that for a linear system $x'=Ax$, there exists a matrix $Q:=\int\limits_0^\infty B(t)dt$, where $B(t)=e^{tA^T}e^{tA}$, and $V(x) = x^T Q x$, $\frac{dV(x(t))}{dt}=\frac{dx^T}{dt}Qx+x^T Q\frac{dx}{dt}=x^T A^T Qx+x^T QAx$.
I don't see how the right-most side was obtained. Would appreciate some clarification.