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I came across a tricky issue while setting up an optimization problem. Is it possible to differentiate $len^2(f(x)_B)$, the squared length of some function $f(x)$ under some basis $B$?

I know this involves summing the squares of the quantities of each element of the basis used to construct the function, but I'm not sure how to express this fact in a form that could be differentiated.

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    $\begingroup$ Differentiate with respect to ... what? $\endgroup$ – MachineLearner Mar 22 '19 at 6:09
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    $\begingroup$ To $x$, ultimately I want to extend the operation to two variables, but I doubt that will be much harder. $\endgroup$ – user10478 Mar 22 '19 at 6:13
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Use the chain rule:

$$L^2(f(x)) \implies 2L(f(x))\dfrac{dL}{df}\dfrac{df}{dx}$$

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    $\begingroup$ Are you sure that is correct? Should't the length be differentiated with respect to $f$ here? $\endgroup$ – Devashsih Kaushik Mar 22 '19 at 6:22
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    $\begingroup$ @DevashishKaushik Thank you for the comment! $\endgroup$ – MachineLearner Mar 22 '19 at 6:24
  • $\begingroup$ No problem ! :) $\endgroup$ – Devashsih Kaushik Mar 22 '19 at 6:29
  • $\begingroup$ @MachineLearner My trouble is with the $\frac{dL}{df}$ factor. How would I expend that part? $\endgroup$ – user10478 Mar 22 '19 at 16:14
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    $\begingroup$ Eh, I think the problem is that I've asked the wrong question entirely. I'll accept this answer and ask a new one with more context. $\endgroup$ – user10478 Mar 22 '19 at 17:02

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