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\end{eqnarray}   Applying Equation \ref{eqn:eqn1} with $f(x) = \|x\|_{\ell_2}^2$, we have   \begin{align*} \begin{equation}  \frac{\partial \|x\|_{\ell_2}^2}{\partial x_i}(x) x_i}  = \frac{\partial \|x\|_{\ell_2}^2}{\partial a_i}(x) a_i}  + j\frac{\partial \|x\|_{\ell_2}^2}{\partial b_i}(x).  \end{align*} b_i}.  \end{equation}