# Derivative quadratic form for regression

I am interested in multilinear regression for Student distribution.

Let $$\mu_i=X_i\beta,$$ to compute estimators for multivariate Student distribution I need to compute the following derivative

$$\frac{\partial \bigl((y_i-x_i\beta)^{T}\Sigma^{-1}(y_i-x_i\beta)\bigr)}{ \partial \beta}.$$

I write $$Q(y_i;x_i\beta,\Sigma)=y_i^T\Sigma^{-1}y_i-2 \beta^T x_i^T\Sigma^{-1}y_i+\beta^Tx_i^T\Sigma^{-1}x_i\beta.$$

Now, if i am not mistaken the derivative of $$\beta^T x_i^T\Sigma^{-1}y_i$$ respect to $$\beta$$ is $$x_i^T\Sigma^{-1}y_i$$.

I am not sure how can I compute the derivative of $$\beta^Tx_i^T\Sigma^{-1}x_i\beta\quad ?$$

$\beta ^T x ^T \Sigma^{-1} x \beta =(\Sigma ^{-1/2}x \beta)^T\Sigma ^{-1/2}x \beta=\|\Sigma ^{-1/2}x \beta\|_2^2$, taking derivative w.r.t. $\beta$ of the square of the norm using the chain rule. yields $$2x^T \Sigma^{-1/2}(\Sigma^{-1/2}x\beta) = 2x^T\Sigma^{-1}x\beta .$$