# Derivative of squared form.

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 $$x_{ji} = (x_{j1}, \ldots, x_{jn})$$.

• 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.
• $\Sigma$ is symmetric. But I don't know how to apply the chain rule in this case. Without the $g$ function I know. Oct 30, 2018 at 23:23