For a differentiable function $f(\bold x)$ and differentiable function $g(y, \bold x)$ with continuous derivatives:

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

A useful way to think about this identity is to compare $f(\bold x)$ to a vector with infinte elements whose value is indexed by $\bold x$. Then, this could be compared to the discrete definition for a vector $\bold \theta$:

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

Important Functional Derivative Identity

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

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

Functionals

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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