multilinear transformation and derivatives question

Question

Let $B ∈ \mathcal L^2(\mathbb R^n, \mathbb R^m)$ and $f (\mathbf x) = B(\mathbf x, \mathbf x)$. Show that $D f (\mathbf x_0)(\mathbf h) = B(\mathbf x_0, \mathbf h) + B(\mathbf h, \mathbf x_0)$.

Answer 1

Note that $f=B\circ (id,id)$, so $Df(x) = DB(id(x),id(x)) \circ (D(id)(x), D(id)(x)) = DB(x,x) \circ (id,id)$.

Thus, $Df(x)h = DB(x,x)(h,h)$.

Recall that $DB(x,y)(h,k) = B(x,k) + B(h,y)$.