Generalization of the notion of derivative to functionals, i.e., to functions that take other functions as an argument. Functional derivatives are particularly useful, for example, in field theory.
Usually in physics one is concerned with functions that take a number of a vector as an input and has another number or vector as an output. to differentiate these functions, one can simply employ the usual techniques from differential calculus.
However, there are some situations in which one wants to consider different sorts of mathematical objects. For example, the action is a functional: a function that takes another function as input. For example, in Classical Mechanics one has $$S[q] = \int_{t_0}^{t_1} L(q(t), \dot{q}(t), t) \ \text{d}t,$$ where $L$ is the system's Lagrangian.
In these situations, one might be interested in differentiating the functional $S$ with respect to its argument, $q$. As an example, Hamilton's principle is equivalent to $\frac{\delta S}{\delta q} = 0$. This notion of derivative is known as a functional derivative: the derivative of a functional with respect to its argument.
