I can't convince myself whether the following functional derivative is trivial or not:(adsbygoogle = window.adsbygoogle || []).push({});

##\frac \delta {\delta \psi(x)} \big[ \partial_x \psi(x)\big],##

where ##\partial_x## is a standard derivative with respect to ##x##.

One could argue that

## \partial_x \psi(x) = \int dx' [\partial_{x'} \psi(x')] \delta (x - x') = - \int dx' \psi(x') \partial_{x'} \delta (x - x'),##

assuming there is no boundary term in integration by parts.

In this case, the functional derivative would give

##\frac \delta {\delta \psi(x)}\Big[ - \int dx' \psi(x') \partial_{x'} \delta (x - x') \Big] = - \partial_{x'} \delta (x - x')\Big|_{x'=x} = 0.##

Any thoughts? Is this rigorous?

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# Functional derivative of normal function

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