I need to compute:
$$ \frac{d}{dx(t)}\int_0^Tx(\tau)^TAx(\tau)d\tau, $$
where $t\in (0,T)$, $A\in\mathbb R^{n\times n}$ and $x\in\mathbb R^n$. Using Leibniz differentiation under an integral sign, I have:
$$ \frac{d}{dx(t)}\int_0^Tx(\tau)^TAx(\tau)d\tau = \int_0^T\frac{dx(\tau)^TAx(\tau)}{dx(t)}d\tau, $$
My course notes say this equals $Ax(t)$, which is almost as if:
$$ \int_0^T\frac{dx(\tau)^TAx(\tau)}{dx(t)}d\tau = \int_0^TAx(\tau)\delta(\tau-t)d\tau $$
But I am not sure whether this is true or if something else tells us that it is $=Ax(t)$. Thanks for helping!