Derivative of $\|Xa\|_2 $ with respect to $X$ Can someone give me the answer to the following expression?

$\frac{\partial}{\partial X}\|Xa\|_2 =?$

$X$ is a square matrix and $a$ is a vektor of the apropriate size. $\|\cdot\|_2$ is the euclidean norm.
Thanks
Take any matrix $B$ and expand 
$$
\|(X+tB)a\|_2^2 = \|Xa\|_2^2 + 2 t \langle Xa , Ba \rangle + t^2 \|Ba\|_2^2.
$$
Therefore the directional derivative of the square of your map in the direction $B$ is
$$
2 B^t Xa
$$
where $B^t$ is the transposed matrix of $B$.
If the following is correct, then how are the above answers related to it?

$$
\frac{\partial}{\partial X}\left[\|Xa\|_2\right] = \frac{\partial}{\partial X}\left[(a^TX^TXa)^{\frac{1}{2}}\right]
= \frac{1}{2}(a^TX^TXa)^{-\frac{1}{2}}\cdot\frac{\partial}{\partial X}\left[(a^TX^TXa)\right]
=\frac{1}{2}(a^TX^TXa)^{-\frac{1}{2}}X(aa^T + aa^T)
= (a^TX^TXa)^{-\frac{1}{2}}Xaa^T
= \frac{Xaa^T}{\|Xa\|_2}
$$