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 $\bold \theta$:

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