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

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

$$F$$ 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 $$F$$.