Let $n \geq 1$. Calculate the differential of the function G:  

$$G: M\_n( \mathbb{R}) \times M\_n( \mathbb{R}) \rightarrow M\_n( \mathbb{R})$$ $$(A,B)\rightarrow A^tBA$$

My first thought was to add small matrix variations $H$ and $K$ to $A$ and $B$ and then try to write as in the definition: $$D\_{(A,B)}G=A^tBA + Lin(H,K)+o(||(H,K)||)$$  

But then I end up with a long string of terms in $H,K$ and I can't determine which are the ones that fall into the Landau $o(||(H,K)||)$ and which are part of the linear expression. 

So, could anyone explain to me where "to draw the line"?