# Product rule for bilinear maps between Euclidean spaces

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 $$dF$$?

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