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 $\boldsymbol{\theta}$:

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