Is the change in a derivative the same as the derivative of the change in a field $\phi$, i.e.
$$\delta(\partial_\mu\phi) = \partial_\mu(\delta \phi)$$
This came up in a Proof of Noether's theorem, when computing the change in the Lagrangian $\delta L$, where $L$ is a function of $\phi$ and its derivates $\partial_\mu \phi$.