In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional (a functional in this sense is a function that acts on functions) to a change in a function on which the functional depends.

In the calculus of variations, functionals are usually expressed in terms of an integral of functions, their arguments, and their derivatives. In an integrand L of a functional, if a function f is varied by adding to it another function δf that is arbitrarily small, and the resulting integrand is expanded in powers of δf, the coefficient of δf in the first order term is called the functional derivative.

For example, consider the functional $${\displaystyle J[f]=\int _{a}^{b}L(\,x,f(x),f'{(x)}\,)\,dx\,,}$$ where f ′(x) ≡ df/dx. If f is varied by adding to it a function δf, and the resulting integrand L(x, f +δf, f ′+δf ′) is expanded in powers of δf, then the change in the value of J to first order in δf can be expressed as follows: $${\displaystyle {\begin{aligned}\delta J&=\int _{a}^{b}\left({\frac {\partial L}{\partial f}}\delta f(x)+{\frac {\partial L}{\partial f'}}{\frac {d}{dx}}\delta f(x)\right)\,dx\,\\[1ex]&=\int _{a}^{b}\left({\frac {\partial L}{\partial f}}-{\frac {d}{dx}}{\frac {\partial L}{\partial f'}}\right)\delta f(x)\,dx\,+\,{\frac {\partial L}{\partial f'}}(b)\delta f(b)\,-\,{\frac {\partial L}{\partial f'}}(a)\delta f(a)\end{aligned}}}$$ where the variation in the derivative, δf ′ was rewritten as the derivative of the variation (δf) ′, and integration by parts was used in these derivatives.

In this section, the functional differential (or variation or first variation) is defined. Then the functional derivative is defined in terms of the functional differential.

Suppose ${\displaystyle B}$ is a Banach space and ${\displaystyle F}$ is a functional defined on ${\displaystyle B}$. The differential of ${\displaystyle F}$ at a point ${\displaystyle \rho \in B}$ is the linear functional ${\displaystyle \delta F[\rho ,\cdot ]}$ on ${\displaystyle B}$ defined by the condition that, for all ${\displaystyle \phi \in B}$, $${\displaystyle F[\rho +\phi ]-F[\rho ]=\delta F[\rho ;\phi ]+\varepsilon \left\|\phi \right\|}$$ where ${\displaystyle \varepsilon }$ is a real number that depends on ${\displaystyle \|\phi \|}$ in such a way that ${\displaystyle \varepsilon \to 0}$ as ${\displaystyle \|\phi \|\to 0}$. This means that ${\displaystyle \delta F[\rho ,\cdot ]}$ is the Fréchet derivative of ${\displaystyle F}$ at ${\displaystyle \rho }$.

However, this notion of functional differential is so strong it may not exist, and in those cases a weaker notion, like the Gateaux derivative is preferred. In many practical cases, the functional differential is defined as the directional derivative $${\displaystyle {\begin{aligned}\delta F[\rho ,\phi ]&=\lim _{\varepsilon \to 0}{\frac {F[\rho +\varepsilon \phi ]-F[\rho ]}{\varepsilon }}\\[1ex]&=\left[{\frac {d}{d\varepsilon }}F[\rho +\varepsilon \phi ]\right]_{\varepsilon =0}.\end{aligned}}}$$ Note that this notion of the functional differential can even be defined without a norm.

In a more general case the function space ${\displaystyle B}$ appearing as the domain of ${\displaystyle F}$ is not a vector space, and therefore variations of the form ${\displaystyle \rho +\varepsilon \phi }$ do not make sense. In this case we consider a variation ${\displaystyle \alpha _{?}:(-\varepsilon _{0},\varepsilon _{0})\to B}$ of ${\displaystyle \rho }$ to be a ${\displaystyle C^{1}}$-family of functions such that ${\displaystyle \alpha _{0}=\rho }$. Denoting the space of all such variations as ${\displaystyle {\mathcal {V}}_{\rho }}$, the functional differential ${\displaystyle \delta F[\rho ]:{\mathcal {V}}_{\rho }\to \mathbb {R} }$ is the functional $${\displaystyle {\begin{aligned}\delta F[\rho ;\alpha ]=\delta F[\rho ][\alpha ]=\lim _{\epsilon \to 0}{\frac {F[\alpha _{\epsilon }]-F[\rho ]}{\epsilon }}=F[\alpha _{?}]'(0)\end{aligned}}}$$

where ${\displaystyle F[\alpha _{?}](\epsilon )=F[\alpha _{\epsilon }]}$. The above then becomes the special case ${\displaystyle \alpha _{\epsilon }=\rho +\epsilon \eta }$.