It's a standard result from matrix calculus that for $X \in \mathbb{R}^{b \,\times\, c}$, we have $$ \frac{\partial \ln\det(X^t X)}{\partial X} = 2(X^\dagger)^t, $$
where $X^t$ denotes matrix transpose, and $X^\dagger$ the pseudoinverse.
I'm currently faced with the following problem, and I don't have enough knowledge of matrix calculus to know how to proceed. Let $f_\theta : \mathbb{R}^a \to \mathbb{R}^{b \,\times\, c}$ be parameterized by a vector $\theta \in \mathbb{R}^n$, and let $x \in \mathbb{R}^a$. I would like to compute $$ \frac{\partial \ln\det(f_\theta(x)^t \, f_\theta(x))}{\partial \theta}. $$
From the identity above, I'm guessing that the result will take the form $$ 2 (f_\theta(x)^\dagger)^t \;\square\; (D_\theta f_\theta)(x), $$ where $D_\theta$ denotes differentiation with respect to $\theta$. But I don't know what $\square$ should be, since the first term is a $b \times c$ matrix, and the second term a $b \times c \times n$ tensor.
Thanks for your help!