I have a little problem with notation and I do not know how to work it out.
I have the next system
$$ \dot x = A(x)x, $$ where $x \in \mathbb{R}^n$ and the square matrix $A(x)_{n\times n}$ has elements depending on some of the components of $x$.
If I want to compute the Jacobian of this system about $x=0$ (zero vector), in order to study the stability of such point, how can I do it in a compact form? I mean, applying the chain rule (can I do that?) I have
$$ J=\frac{\partial A(x)}{\partial x}x+A(x)\frac{\partial x}{\partial x}, $$
the second term has meaning for me, since $\frac{\partial x}{\partial x}$ is the identity matrix, so the second term at $x=0$ corresponds to $A(0)$. But what about the first term? I do not know how to "write/read or compute" $\frac{\partial A(x)}{\partial x}$ since $x$ is a vector. Is there any way to compute/write down $J$ in a "straight forward" way? With straight forward I mean, using the notation $A(x)$ (as for the second term, which result is $A(0)$) instead of splitting all the equations of the system just taking the rows of $A(x)$.
Thanks in advance