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?


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...

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

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