# Show that the following function is differentiable

I'm trying to solve the following question:

Let $$U$$ be a open set in $$\mathbb{R}^n$$ and $$A: U \rightarrow \mathcal{L}(\mathbb{R}^n,\mathbb{R}^n)$$ a differentiable aplication, with $$\mathcal{L}(\mathbb{R}^n, \mathbb{R}^n)$$ being the space of all linear applications $$T: \mathbb{R}^n \rightarrow \mathbb{R}^n.$$ Show that $$\phi: U \rightarrow \mathbb{R}$$ defined by $$\phi(x) = \left \langle A(x)x, x\right \rangle$$ is differentiable and find $$\phi'(x).$$

One of my main problems is that $$A$$ is very much not a linear aplication, and all the tecniques that I know about differentiation seems to fail on this problem. I'm thinking that if I show, somehow, that $$\phi$$ is a composition of differentiable function, it may work, but I can't think of any combination that does the job. Any ideas on how to proceed?

## 1 Answer

Hints: the evaluation $$\mathcal{L}(\mathbb{R}^n,\mathbb{R}^n)\times\mathbb{R}^n \longrightarrow\mathbb{R}^n$$ $$(T,x)\longmapsto Tx$$ and the scalar product $$\mathbb{R}^n\times\mathbb{R}^n\longrightarrow\mathbb{R}$$ $$(x,y)\longmapsto⟨x,y⟩$$ are bilinear, so...

• I'm sorry, but I didn't get how this helps. Could you please elaborate a little more? Aug 1 '19 at 18:52
• @BrunoTassone, bilinear functions are differentiable. See the link. You can wtrite your function as a composition of differentiable functions. Aug 1 '19 at 20:08