In mathematics, and theoretical physics, the functional derivative is a generalization of the usual derivative that arises in the calculus of variations. In a functional derivative, instead of differentiating a function with respect to a variable, one differentiates a functional with respect to a function.
Two possible, restricted definitions suitable for certain computations are given here. There are more general definitions of functional derivatives.
If we have a functional F mapping (continuous/smooth/with certain boundary conditions/etc.) functions from a manifold M to or , to or , then, provided the following derivative exists, the functional derivative
is a distribution such that for all test functions f,
Another definition is in terms of a limit and the Dirac delta function, :
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