For a differentiable function $$f(\mathbf{x})$$ and a differentiable function $$g(y, \mathbf{x})$$ with continuous derivatives, the functional derivative of the integral of $$g(f(\mathbf{x}), \mathbf{x})$$ is given by:

$$\frac{\partial}{\partial f(\mathbf{x})} \int g(f(\mathbf{x}), \mathbf{x}) d\mathbf{x} = \frac{\partial}{\partial y} g(f(\mathbf{x}), \mathbf{x})$$

A useful way to think about this identity is to compare $$f(\mathbf{x})$$ to a vector with infinite elements whose values are indexed by $$\mathbf{x}$$. Under this analogy, the identity can be compared to the discrete partial derivative for a vector $$\boldsymbol{\theta}$$:

$$\frac{\partial}{\partial \theta_i} \sum_j g(\theta_j, j) = \frac{\partial}{\partial \theta_i} g(\theta_i, i)$$

Functional Derivative Identity for Integrals

Also known as a **variational derivative**, the functional derivative of a functional $$J[f]$$ is denoted $$\frac{\partial}{\partial f(\mathbf{x})} J$$, and is defined as the derivative of $$J$$ with respect to the value of the function $$f$$ at the point $$\mathbf{x}$$.

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A function of a function $$f$$ is called a **functional** $$J[f]$$.

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Goodfellow, I., Bengio, Y., & Courville, A. (2016). $\mathit{Deep \ Learning.}$ MIT Press. Retrieved from [www.deeplearningbook.org](https://www.deeplearningbook.org) 

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