What is the derivative of the function involving inner product $f(x)=\langle Ax,Bx\rangle$

Question

I want to find the first and second derivative of $f(x)=\langle Ax,Bx\rangle$, where $\langle\,,\rangle$ is the inner product, A and B are $n\times n$ matrices and $x\in \mathbb{R}^n$.
Here's my thoughts:
Since A and B are linear transformations, their derivatives are themselves. So maybe we can treat f as a function from $\mathbb{R}^n\times \mathbb{R}^n$ to $\mathbb{R}$?

Answer 1

As the inner product is defined by $\langle u, v\rangle = u^Tv$ we can reformulate your function to $f(x) = (Ax)^TBx = x^TA^TBx$

As seen here: Differentiate $f(x)=x^TAx$ you get the derivative $$f'(x) = 2x^TA^TB$$ Calculating the next derivative should be much easier now.