Jacobian for a composition of two bilinear functions

Question

Can somebody tell me what the jacobian of $f(x) = \frac{x^T C x}{a^2 + x^T C x}$ looks like and how to get there? I was thinking about using $f(x) = g(x) \cdot h(x)^{-1}$ and using the product rule, is this feasible? Thanks!

Answer 1

you can apply the formulas $J=\nabla (F(x))^T$ and $\nabla \left(\frac{f}{g}\right)=\frac{g\nabla f-f\nabla g}{g^2}$

(see https://math.stackexchange.com/questions/84518/quotient-rule-extendable-to-functions-of-vectors}

so if $B=x^TCx$, you will get $J=\left(\frac{(a^2\nabla B)}{(a^2+B)^2}\right)^T$