What is the functional derivative with respect to a derivative?

Question

Does the following functional derivative can be evaluated?
$$ frac{partial}{partial(partial_mu phi(x))} int d^4y F(y) partial_nupartial^nuphi(y)$$
I am trying to find the equations of motion of a classical field with a Lagrangian that has the form:
$$mathcal{L}(phi,partial_muphi)+int d^4y F(y) partial_nupartial^nuphi(y)$$
Is the conventional form of E-L equation valid here:
$$ frac{partial mathcal{L}}{partial(partial_mu phi(x))}- frac{partial mathcal{L}}{partial phi}=0$$
Or is a new form necessary due to the dependence on the second derivative inside the integral of the Lagrangian?

drfk · Answer

You can just partially integrate the double derivative to get $-partial_nu F partial^nu Phi$. Or use a definition for the functional derivative analog to the usual derivative, where you'd, in this case, replace $partial^nu Phi(y)$ by the delta-distribution of $delta(x-y)$.

Answered by drfk on October 6, 2020

What is the functional derivative with respect to a derivative?

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