Derivative with respect to a functional

Solution 1:

define the linearization of the functional $F$ by $F[f+\delta f]=F[f]+F_1\circ\delta f+{\cal O}(\delta f)^2$, where $F_1$ is a linear functional; denote the differential operator by $D$; then $\delta F[f']=F_1\circ D\circ\delta f$, $\delta F[f]=F_1\circ\delta f=F_1\circ D^{-1}\circ F_1^{-1}\circ\delta F\circ f'$, and hence $$\frac{\delta F[f]}{\delta F[f']}=F_1\circ D^{-1}\circ F_1^{-1}.$$

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