Differentiation rule for $A(x)x$ with $x$ I know that the differentiation rule for $Ax$ is as follows
$d(Ax)/dx$ = $x$ where $A$ is a matrix and $x$ is a vector.
But what is the differentiation rule for $A(x)x$ with $x$.
$A$ is a function of vector $x$
By my intuition is seems like below
$d(A(x)x)/dx = dA(x)/dx*x + A $
Could somebody help with to verify if my intuition is correct and can provide proof if is correct for the same(if it is correct).

That seems right, but you need to take care: you are taking derivative of a vector as a function of a vector. This means the result is a matrix.
There is some subtlety with the notation $\frac{\mathrm dA}{\mathrm dx}$. Using indices, $A_{ij}$ is the $i$-th row, $j$-th column of the matrix $A$. Then $$(\frac{\mathrm dA}{\mathrm dx})_{ijk} = \frac{\mathrm{d}A_{ij}}{\mathrm{d}x_k}$$ has three indices! This is best described with what is called tensors.
Refer to your textbook for more information, or ask a new question about this.
