Why was the higher order derivative defined this way?

[cliowa]

Let E, F be Banach spaces, and let $L(E;F)$ denote the space of linear, bounded maps between E and F. My goal is to understand better higher order derivatives.
Let's take $E=\mathbb{R}^2, F=\mathbb{R}$. Consider a function $f:U\subset\mathbb{R}^2\rightarrow\mathbb{R}$, where U is an open subset of $\mathbb{R}^2$. Then $D^2 f:U\rightarrow L(\mathbb{R}^2;L(\mathbb{R}^2;\mathbb{R}))$.
Now, I read that for $u\in U, v,w\in\mathbb{R}^2$ by definition $D^2 f(u)\cdot (v,w):=D((Df)(.)\cdot w)\cdot v$. My question now is: Why was this defined precisely this way?
Does it have something to do with "using the product rule", which would amount to $D((Df)(.)\cdot w)=D^2 f(.)\cdot w+Df(.)\cdot D(w)=D^2 f(.)\cdot w$?
Thanks for any help. Best regards...Cliowa

 

[fresh_42]

It has to do with the definition of derivatives. $D^2(f)= D(D(f))$. We get a bilinear functional from that. How to consider a derivative is a matter of purpose. E.g. look at the ten point list at the beginning of
https://www.physicsforums.com/insights/journey-manifold-su2mathbbc-part/where I listed a few of such perspectives. Your view is: $D$ is a linear functional, and $D^2$ a bilinear. Look up differential forms. E.g.
https://www.physicsforums.com/insights/the-pantheon-of-derivatives-i/