Total derivative of the function $F(x,y)=\langle Ax,y \rangle$, where $A$ is a matrix 
Let $F:\mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}$. Defined as $F(x,y)=\langle Ax,y \rangle$ where $A$ is an $n \times n$ matrix. Then show that $$(DF(x,y))(u,v)=\langle Au,y \rangle +\langle Ax,v \rangle,$$ where $D $ is the total derivative.

We have just been introduced to multivariate calculus this semester. I have been trying to crack this one since morning. I am sure there is a key idea that I am missing.

If $B:\mathbb R^n\times \mathbb R^n\longrightarrow \mathbb R$ is bilinear, then $DB(a,b)(h,k) = B(a,k) + B(h,b)$. Indeed, note that
$$B(a+h,b+k)-B(a,b)-B(a,k)-B(h,b)=B(h,k)$$
and $B(h,k)$ is of quadratic order, so $B(h,k)/\lVert (h,k)\rVert \to 0$. 
