Expected Value of $x_1 \exp(a_1 x_1 + a_2 x_2 ... a_n x_n)$ when X is multivariate $N(0, \Sigma)$ Which is the
$E(x_1 \exp(a_1 x_1 + a_2 x_2 + \dotsm + a_n x_n)$ when X is an n-random vector distributed multivariate normal (0, $\Sigma$).

Letting
$$
M_{\mathbf{x}}(\mathbf{t})=Ee^{\mathbf{t}^T\mathbf{x}}=e^{\mathbf{t}^T\mathbf{\Sigma}\mathbf{t}/2}
$$
denote the moment generating function of $\mathbf{x}$ it follows that
\begin{align}
E\left(x_1e^{\mathbf{a}^T\mathbf{x}}\right)
&=E\left(\frac{\partial}{\partial a_1}e^{\mathbf{a}^T\mathbf{x}}\right)
\\&=\frac{\partial}{\partial a_1}Ee^{\mathbf{a}^T\mathbf{x}}
\\&=\frac{\partial}{\partial a_1}M_{\mathbf{x}}(\mathbf{a})
\\&=\frac{\partial}{\partial a_1}e^{\mathbf{a}^T\mathbf{\Sigma}\mathbf{a}/2}
\\&=e^{\mathbf{a}^T\mathbf{\Sigma}\mathbf{a}/2}\mathbf{\Sigma}_{1\cdot}{\mathbf{a}}
\end{align}
where $\mathbf{\Sigma}_{1\cdot}$ denotes the first row of $\mathbf{\Sigma}$.
