Derivative of squared form.

Question

I'd like to calculate the derivative $\frac{\partial u}{\partial \beta}$, where

$$ u = [y - g^{-1}(X^T\beta)]^T \Sigma^{-1}[y - g^{-1}(X^T\beta)], $$

$y = (y_1, \ldots, y_n)^T$ and $\beta = (\beta_1, \ldots, \beta_p)^T$ are vectors and $x_{ji} = (x_{j1}, \ldots, x_{jn})$.

Answer 1

Hints: (use chain rule, the following facts may help)

• For symmetric matrix $S$, we have $\nabla_\beta (\beta^\top S \beta) = 2 S \beta$.
• For a matrix $A$, we have $\nabla_\beta (A \beta) = A$.
• You will have to take the derivative of $g^{-1}$ at some point.