Show that $Df(V,W)$ evaluated on $(H,K)$ is given by $f(V,K)+f(H,W)$. Let $f:\Bbb R^2\times \Bbb R^2\to \Bbb R$ be a bilinear map.Then show that for $(V,W)\in \Bbb R^2\times \Bbb R^2$,the derivative $Df(V,W)$ evaluated on $(H,K)\in \Bbb R^2\times \Bbb R^2$ is given by 
$f(V,K)+f(H,W)$.
As I could not understand how to prove it ;I took an example first.
Let $f:\Bbb R^2\times \Bbb R^2\to \Bbb R$ be defined by $f((x_1,y_1)+(x_2,y_2))=(x_1+2y_1+x_2+2y_2)$ where $V=(x_1,y_1),W=(x_2,y_2)$,
$Df(V,W)=\begin{bmatrix} 1,2,1,2\end{bmatrix}$.
where I did the derivative at each component.
How how will it equal $f(V,K)+f(H,W)$?I am confused totally.Please help someone

What is the definition of the derivative of a bilinear form? Is it a generalization of Fréchet derivative? Then we should just look for linear terms in the 
$$f(v+h,w+k)-f(v,w),$$
which after discarding the second-order term $f(h,k)$ leads to
$$[Df(v, w)](h, k) = f(v, k) + f(h, w).$$
