Let $$F:\Bbb R^n\times\Bbb R^n\to\Bbb R$$ be the function $F(x,y)=\langle Ax,y\rangle$ where $\langle , \rangle$ denotes the standard inner product on $\Bbb R^n$ and $A$ be an $n\times n$ real matrix. If $D$ denotes the total derivative. which of the fallowing is correct?
1). $(DF(x,y))(u,v)=\langle Au,y\rangle+\langle Ax,v\rangle$
3). $DF(x,y)$ may not exist for some $(x,y)\in\Bbb R^n\times\Bbb R^n$
4). $DF(x,y)$ doesnot exist at $(x,y)=(0,0)$
I don't know, how the total derivative is defined for Inner product. So, tell me the definition for total derivative on inner product space, also tell, How to proceed further to solve this problem? Thank you