Suppose there are two positive definite quadratic forms $f(\mathbf x) = \mathbf x ^ T \mathbf C \mathbf x $ and $g(\mathbf x) = \mathbf x ^ T \mathbf G \mathbf x $. My goal is to get an expression for the derivative $$ \frac{\mathrm d}{\mathrm d \mathbf x} \left ( \frac {f(\mathbf x)}{g(\mathbf x)} \right ) .$$
Allegedly, $$ \frac{\mathrm d}{\mathrm d \mathbf x} \left ( \frac {f(\mathbf x)}{g(\mathbf x)} \right ) = \frac{\mathbf C \mathbf x f(\mathbf x) - \mathbf G \mathbf x f(\mathbf x)}{g(\mathbf x)^2} ,$$ although I don't know how to justify this result, or whether it is infact correct. Can somebody please show how to take this derivative. Please show a few intermediate steps, to facilitate understanding. Please do use $ \frac{\mathrm d}{\mathrm d \mathbf u} \left ( \mathbf u ^ T \mathbf A \mathbf u \right ) = 2 \mathbf u ^ T \mathbf A$ if appropriate.