Second derivative of a matrix quartic form I need to compute the second derivative of the following quartic expression: $$x^H A^H x x^H A x$$ where is Hermitian. I have tried to compute the first derivative, and if I am not wrong, it should be: $$(A+A^H) x x^H (A+A^H) x$$
But then, I do not know how to proceed to calculate the second derivative. Could someone sketch the steps I need to follow? Thank you.

The second derivative of a multi-variable function $f=f(x_1,\ldots,x_n)$ is usually expressed as its Hessian matrix
$$
Hf(x_1,\ldots,x_n)=\left(\frac{\partial^2 f}{\partial x_i\partial x_j}\right)_{i,j=1}^n
$$
Here
$$
f(x)=x^tA^txx^tAx=(x^tBx)^2, \quad \text{where $B=\frac{1}{2}(A+A^t)=(b_{ii})$}.
$$
Hence
$$
\frac{\partial f}{\partial x_i}=4(x^tBx)(Bx)_i
$$
and 
$$
\frac{\partial^2 f}{\partial x_i\partial x_j}=8(Bx)_i(Bx)_j+4(x^tBx)b_{ij},
$$
and thus
$$
Hf(x)=8Bx(Bx)^t+4(x^tBx)B
$$
