If $F(x,y) = \langle Ax,y \rangle$, find derivative of $F(x,y)$

Question

Let $A$ be a $n \times n$ real matrix and $f:\Bbb{R}^n \times \Bbb{R}^n \to \Bbb{R}$ such that $ f(x,y)= \langle Ax,y \rangle$, where $ \langle x,y \rangle$ denotes the inner product of $x$ and $y$. Let $Df(x,y)$ denote the derivative of $f$ at $(x,y)$ which is a linear transformation from $\Bbb{R}^n \times \Bbb{R}^n \to \Bbb{R}$.

My question is how to find $Df(x,y)?$ I tried to use jacobian but I cannot.Please give me a hint to solve

Answer 1

$F$ is bilinear, so $DF_{(x,y)}(u,v)=\langle A(x),v\rangle+\langle A(u),y\rangle$.

A way to see this is to look at the partial derivative relatively to $x$ and $y$ of $F$.