Krdytkyu

Suppose I have the following expression:

$$dot{mathbf{x}}^intercalleft(mathbf{A}-mathbf{B}mathbf{D}^{-1}mathbf{B}^intercalright)dot{mathbf{x}}$$

where $mathbf{x}(t)in mathbb{R}^{m}$ is a vector and $mathbf{A}in mathbb{R}^{mtimes m}, mathbf{B}in mathbb{R}^{mtimes (m-n)}$ and $mathbf{D}in mathbb{R}^{(m-n)times(m-n)}$ are matrices that are dependent on $mathbf{x}(t)$.

Derivation of the expression with respect to $dot{mathbf{x}}$ yields

$$2dot{mathbf{x}}^intercalleft(mathbf{A}-mathbf{B}mathbf{D}^{-1}mathbf{B}^intercalright)$$

according to the rules discussed in this Wikipedia article about matrix calculus.

What happens if I would like to further derive this expression w.r.t. time?

So my question in short: What does this expression equal to?

$$dfrac{text{d}}{text{d}t}left(dfrac{partial}{partial dot{mathbf{x}}}left(dot{mathbf{x}}^intercalleft(mathbf{A}(mathbf{x}(t))-mathbf{B}(mathbf{x}(t))mathbf{D}^{-1}(mathbf{x}(t))mathbf{B}^intercal(mathbf{x}(t))right)dot{mathbf{x}}right)right)$$

Thank you in advance!

Define the variables
$$eqalign{
v &= {dot x},,,,,,M&=A-BD^{-1}B^T cr
{mathcal E} &= v^TMv &= M:vv^T cr
}$$
where colon is a convenient product notation for the trace, i.e. $,,A:B={rm Tr}(A^TB)$

Now calculate the differential and gradient of the scalar function
$$eqalign{
d{mathcal E}
&= M:(dv,v^T+v,dv^T) cr
&= (M+M^T):(dv,v^T) cr
&= (M+M^T)v:dv cr
p=frac{partial{mathcal E}}{partial v} &= (M+M^T)v cr
}$$
NB:   If $M=M^T$ we can simplify this to $,,p = 2Mv$.

Now take the time derivative of the $p$ vector
$$eqalign{
{dot p} &= 2{dot M}v + 2M{dot v} cr
}$$
Since you were vague on how your matrices depend on the $x$-vector, that's about as far as I can take it.