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and it follows that  \begin{equation}  \frac{\partial (Ax)}{\partial x_1} = \left[ \begin{array}{c}  \frac{\partial (Ax)_1}{\partial x_1} \\ \vdots \\ \frac{\partial (Ax)_m}{\partial x_m} x_1}  \end{array} \right]  = \left[ \begin{array}{c} A_{11} \\ \vdots \\ A_{m1} \end{array} \right].  \end{equation}