I'm trying to solve below exercise
Let $(H, \langle \cdot, \cdot \rangle)$ be a real Hilbert space. Let $a: H \times H \rightarrow \mathbb{R}$ be a continuous bilinear form. Determine the derivative of $F: H \to \mathbb R, v \mapsto a(v, v)$.
Could you confirm if I correctly apply below Lemma and the chain rule?
Lemma Let $E_1, \ldots, E_m, F$ be Banach spaces over the field $\mathbb{K} \in \{\mathbb{R}, \mathbb{C}\}$. Let $\varphi: E_1 \times \cdots \times E_m \to F$ be a continuous multilinear map. Then $\varphi$ is continuously differentiable with $\partial \varphi (x_1, \ldots, x_m) : E_1 \times \cdots \times E_m \to F$ such that $$ \partial \varphi (x_1, \ldots, x_m) [h_1, \ldots, h_m] = \sum_{j=1}^m \varphi\left(x_1, \ldots, x_{j-1}, h_j, x_{j+1}, \ldots, x_m\right) $$ for every $(h_1, \ldots, h_m) \in E_1 \times \cdots \times E_m$.
My attempt Let $G: H \to H \times H, v\mapsto (v, v)$. Then $G$ is linear continuous. Then $\partial G (v) = G$ for all $v \in H$. We have $F = a \circ G$. By chain rule, $$ \partial F (v) = \partial a(G(v)) \circ \partial G (v) = \partial a(v, v) \circ G. $$
By Lemma, $$ \partial F (v) [u] = a(u, v) + a(v, u). $$