Let $A$ be an $n\times n$ matrix with smooth entries in $x$, such that $A(x)$ is invertible everywhere with smooth entries. Define a function $f\colon \mathbb{R}^n\to\mathbb{R}^n$ by $f (x)=[A (x)^{-1}]^Tb$ for some $b\in \mathbb{R}^n$. Show that $A^TM$ is a symmetric matrix where $M= D f$ is the derivative of $f$ with respect to $x$.
This problem arises in geometry (see page 12 of these notes math.berkeley.edu/~evans/semiclassical.pdf) and I know an indirect way of solving it, but it doesn't help me much in understanding why this is true. Any thoughts would be appreciated.