I have a quadratic form of the form $Q(\Sigma; x, \mu) = (x-\mu)'\Sigma^{-1}(x-\mu)$ where $\Sigma$ is a positive-definite non-singular matrix with (modified) Cholesky decomposition $\Sigma = LDL'$ with $L$ being a lower-triangular matrix (with diagonals all unity).
I want to find the derivative of the above quadratic form with respect to the entries in $L$. That is, I want to find
$$\frac\partial{\partial L} Q(\Sigma;x,\mu).$$
Is this an established result? are there references I could look at?
Many thanks!