Are taking the real part and differentiating associative? Heya all this is a quick question which I think is true but consider a continuous function $f(x): \mathbb{R} \to \mathbb{C} $ then is $\mathbb{R}\{\frac{d}{dx}f(x) \} = \frac{d}{dx}\mathbb{R}\{f(x)\}$
In words:
Is differentiating the real part of a function the same as the real part of the derivative?
Every example I have tried so far it ends up holding!

Yes, because differentiating $f:\mathbb R\to\mathbb C$ is defined as
$$\frac{\mathrm d}{\mathrm dx}f(x) := \frac{\mathrm d}{\mathrm dx}(\mathfrak{Re}f)(x)+i\frac{\mathrm d}{\mathrm dx}(\mathfrak{Im}f)(x)$$
Therefore,
$$\mathfrak{Re}\left(\frac{\mathrm d}{\mathrm dx}f(x)\right) = \frac{\mathrm d}{\mathrm dx}(\mathfrak{Re}f)(x)$$
