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Can someone help explain how differentiating the following with respect to $x$:

$$ \frac{1}{2} \alpha \mathbf{x}^T \Sigma \mathbf{x} + (\mathbf{\mu} - R\mathbf{1})\mathbf{x} $$

Yields the following:

$$ \alpha \Sigma \mathbf{x} + (\mathbf{\mu} - R\mathbf{1}) $$

Where $\Sigma$ is a correlation matrix.

I'm rusty with my linear algebra so the derivate of these transpose matrices isn't making any sense to me. A detailed explanation would be very much appreciated. What happens to the 1/2, and that whole first term in general?

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Could you please be more specific with your question and post the text here? This will be more helpful for other people visiting the site.

Now as far as to where the 1/2 went, usually people put 1/2 in front of the second order term because this will simplify to 1 after the derivation:

$$ \frac{\partial x^2}{\partial x} = 2x $$ vs $$ \frac{1}{2} \cdot \frac{\partial x^2}{\partial x} = x $$

To understand what are the mechanics behind the derivation in your question, I suggest creating your own 2x2 matrices and going through the calculations.

In your case $\Sigma$ is symmetric, in which case:

$$ \frac{\partial \mathbf{x}^T \mathbf{A} \mathbf{x}}{\partial x} = 2\mathbf{A}\mathbf{x} $$ vs

To learn more, go HERE and take a look at the scalar-by-vector identities.

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