Chain rule of derivatives on $x(t)^T A(t) x(t)$

Question

I'm trying to differentiate $\frac{d}{dt}x(t)^T A(t) x(t)$

I know that

$\frac{d}{dx}x^T A x$ is $2Ax$

and

$\frac{d}{dA}x^T A x$ is $xx^T$

according to wikipedia

but I'm not sure how to account for the $x'$ term and $A'$ term that should pop out via the chain rule of derivatives.

My best guess, trying to get compatible shapes, is:

$\frac{d}{dt}x(t)^T A(t) x(t) = 2x'^TAx + A'^Txx^T$, however, this is a scalar plus a matrix. The derivative should just be a scalar.

What is $\frac{d}{dt}x(t)^T A(t) x(t)$?

Answer 1

Using chain rule would be very complicated because of how derivatives work with respect to higher order tensors (e.g. matrices, three-dimensional arrays, etc.). Instead product rule works just fine:

$$\frac{d}{dt}\mathbf{x^TAx} = \mathbf{(x')^TAx} + \mathbf{x^TA'x} + \mathbf{x^TAx'}$$