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I am reading through a book called "Matrix Differential Calculus" by Magnus and Neudecker. They go through taking the derivative of a vector in quadratic form that I need help with.

For [itex] \vec{x} [/itex] being a vector and A being a constant square matrix

[tex] \frac {d(\vec{x}^TA\vec{x})} {d\vec{x}} [/tex]

[tex]= \frac {d\vec{x}^T}{d\vec{x}}A\vec{x} + \vec{x}^TA\frac {d\vec{x}}{d\vec{x}} [/tex]

[tex]= (\frac {d\vec{x}^T}{d\vec{x}}A\vec{x})^T + \vec{x}^TA\frac {d\vec{x}}{d\vec{x}} {?}[/tex]

[tex]= \vec{x}^TA^T\frac {d\vec{x}}{d\vec{x}} + \vec{x}^TA\frac {d\vec{x}}{d\vec{x}} [/tex]

[tex]= \vec{x}^T(A+A^T)\frac {d\vec{x}}{d\vec{x}} [/tex]

[tex] \frac {d(\vec{x}^TA\vec{x})} {d\vec{x}}=\vec{x}^T(A+A^T) [/tex]

I don't understand how one can go from the second line to the third line

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# Derivative of a vector function

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