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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 $\mathbb{R}$?

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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.

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  • $\begingroup$ $A^TB$ is not necessarily symmetric. $\endgroup$ – user10354138 May 10 at 11:51
  • $\begingroup$ Oh you're right, I oversaw that. I didn't find a formula for the derivative of $x^TAx$ to the fast, but I'm sure there is one $\endgroup$ – LegNaiB May 10 at 12:03

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