Derivative of a products with matrix exponential

Question

I should find the derivative with respect to the vector $a \in \mathbb{R}^{n}$

$\dfrac{\partial}{\partial a}(a^{T}exp(aa^{T})a)$

Answer should be in a matrix form

I tried to decompose the expression into a derivative of products, but $\dfrac{\partial a}{\partial a}$ looks strange

Answer 1

Hint :

Tailor's Theorem gives \begin{align*} \exp(aa^T)&=\sum_{n=0}^\infty \frac{1}{n!} (aa^T)^n\\ &=\sum_{n=0}^\infty \frac{1}{n!} a (a^T a)^{n-1} a^T\\ &=a \left( \sum_{n=0}^\infty \frac{1}{n!} (\lVert a \rVert^2)^{n-1} \right) a^T\\ &=\frac{1}{\lVert a \rVert^2} a \left( \sum_{n=0}^\infty \frac{1}{n!} (\lVert a \rVert^2)^{n} \right) a^T\\ &=\frac{\exp(\lVert a \rVert^2)}{\lVert a \rVert^2} aa^T \end{align*}

So that \begin{align*} a^T \exp(aa^T) a &= \lVert a \rVert^2 \exp(\lVert a \rVert^2) \end{align*}