Derivative of Quadratic Form For the Quadratic Form $X^TAX; X\in\mathbb{R}^n, A\in\mathbb{R}^{n \times n}$ (which simplifies to $\Sigma_{i=0}^n\Sigma_{j=0}^nA_{ij}x_ix_j$), I tried to take the derivative wrt. X ($\Delta_X X^TAX$) and ended up with the following:
The $k^{th}$ element of the derivative represented as
$\Delta_{X_k}X^TAX=[\Sigma_{i=1}^n(A_{ik}x_k+A_{ki})x_i] + A_{kk}x_k(1-x_k)$
Does this result look right? Is there an alternative form?
I'm trying to get to the $\mu_0$ of Gaussian Discriminant Analysis by maximizing the log likelihood and I need to take the derivative of a Quadratic form. Either the result I mentioned above is wrong (shouldn't be because I went over my arithmetic several times) or the form I arrived at above is not the terribly useful to my problem (because I'm unable to proceed).
I can give more details about the problem or the steps I put down to arrive at the above result, but I didn't want to clutter to start off. Please let me know if more details are necessary.
Any link to related material is also much appreciated.
 A: You could also take the derivative of the scalar sum.
\begin{equation}
\begin{aligned}
{\bf x^TAx} = \sum\limits_{j=1}^{n}x_j\sum\limits_{i=1}^{n}x_iA_{ji}
\end{aligned}
\end{equation}
The derivative with respect to the $k$-th variable is then(product rule):
\begin{equation}
\begin{aligned}
\frac{d {\bf  x^TAx}}{d x_k} & = \sum\limits_{j=1}^{n}\frac{dx_j}{dx_k}\sum\limits_{i=1}^{n}x_iA_{ji} + \sum\limits_{j=1}^{n}x_j\sum\limits_{i=1}^{n} \frac{dx_i}{dx_k}A_{ji} \\
 & = \sum\limits_{i=1}^{n}x_iA_{ki} + \sum\limits_{j=1}^{n}x_jA_{jk}
\end{aligned}
\end{equation}
If then you arrange these derivatives into a column vector, you get:
\begin{equation}
\begin{aligned}
\begin{bmatrix}
\sum\limits_{i=1}^{n}x_iA_{1i} + \sum\limits_{j=1}^{n}x_jA_{j1} \\
\sum\limits_{i=1}^{n}x_iA_{2i} + \sum\limits_{j=1}^{n}x_jA_{j2} \\
\vdots \\ 
\sum\limits_{i=1}^{n}x_iA_{ni} + \sum\limits_{j=1}^{n}x_jA_{jn} \\
\end{bmatrix} = {\bf Ax} + ({\bf x}^T{\bf A})^T = ({\bf A} + {\bf A}^T){\bf x}
\end{aligned}
\end{equation}
or if you choose to arrange them in a row, then you get:
\begin{equation}
\begin{aligned}
\begin{bmatrix}
\sum\limits_{i=1}^{n}x_iA_{1i} + \sum\limits_{j=1}^{n}x_jA_{j1} &
\sum\limits_{i=1}^{n}x_iA_{2i} + \sum\limits_{j=1}^{n}x_jA_{j2} &
\dots &
\sum\limits_{i=1}^{n}x_iA_{ni} + \sum\limits_{j=1}^{n}x_jA_{jn} 
\end{bmatrix} \\ = ({\bf Ax} + ({\bf x}^T{\bf A})^T)^T = (({\bf A} + {\bf A}^T){\bf x})^T = {\bf x}^T({\bf A} + {\bf A}^T)
\end{aligned}
\end{equation}

A: Let $Q(x) = x^T A x$. Then expanding $Q(x+h)-Q(x)$ and dropping the higher order term, we get $DQ(x)(h) = x^TAh+h^TAx = x^TAh+x^TA^Th = x^T(A+A^T)h$, or more typically, $\frac{\partial Q(x)}{\partial x} = x^T(A+A^T)$.
Notice that the derivative with respect to a column vector is a row vector!
A: It is easier using index notation with Einstein (repeated sum on dummy indices) rule. That is, we can write the $i$th component of $Ax$ as $a_{ij} x_j$, and $x^T A x=x_i a_{ij} x_j
= a_{ij} x_i x_j$. Then take the derivative of $f(\bf{x})$ with respect to a component $x_k$. We find
\begin{eqnarray}
   \partial f/\partial x_k = f,_k = a_{ij} x_{i,k} x_j + a_{ij} x_i x_{j,k} = a_{ij} \delta_{ik} x_j + a_{ij} x_i \delta_{jk} = a_{kj} x_j + a_{ik} x_i, 
\end{eqnarray}
which in matrix notation is $k$th component of ${\bf{x}}^T  A + {\bf{x}}^T A^T$. 
A: Yet another approach using the Frobenius product notation.
For a column vector $x \in \mathbb{R}^n$, and a matrix $A \in \mathbb{R}^{n \times n}$  we can write:
$$ x^TAx = Tr(x^TAx) = x:Ax$$
Then we take the differential and derivative as
\begin{align}
    d(x:Ax) & = dx:Ax + x:Adx\\
            & = Ax:dx + A^Tx:dx\\
            & = (Ax + A^Tx):dx\\
         \frac{\partial (x^TAx)}{\partial x} &= (Ax + A^Tx) = (A + A^T)x 
\end{align}
A: I just learned a new trick when your independent variable is in more than two places within your formula: introduce a new (fake) parameter which will then disappear:
$$\frac{\partial}{\partial x} y^TAx = \frac{\partial y}{\partial x}[Ax]^T+y^TA $$
The transpose was to make the vector a row vector. Nothing deep there!
Now, if $y=x$ then 
$$ \frac{d}{dx} x^TAx = x^TA^T+x^TA = x^T(A+A^T) \ . $$
