Product rule for bilinear maps between Euclidean spaces

Question

Consider

• a bilinear map $\langle \cdot, \cdot\rangle : \mathbb{R}^m \times \mathbb{R}^n \to \mathbb{R}^k$,
• an open set $U \subseteq \mathbb{R}^j$,
• a pair of maps $f: U \to \mathbb{R}^m$ and $g: U \to \mathbb R^n$, and
• the composite map $F(x) = \langle f(x), g(x) \rangle$.

Then, is it necessarily true that $$ dF_{a}(b) = \langle df_{a}(b), g(a) \rangle + \langle f(a), dg_{a}(b) \rangle, $$ and if not, is there a similar product rule for $dF$?

Answer 1

Yes the equality

$$ dF_{a}(b) = \langle df_{a}(b), g(a) \rangle + \langle f(a), dg_{a}(b) \rangle, $$ always holds. This is a consequence of the chain rule as the derivative of the map $G(u,v) = \langle u, v \rangle$ is

$$G^\prime(u,v)(h,k) = \langle u, k \rangle + \langle h, v \rangle.$$