I will begin by stating the question and then I will explain my doubt. The relation between time-derivatives of a vector $\vec{u}$ observed from fixed and rotating frames (with a common origin) is
$$ \left[\frac{d\vec{u}}{dt}\right]_f = \left[\frac{d\vec{u}}{dt}\right]_r + \vec{w} \times \vec{u}$$
Question: I don't understand, in the derivation of this equation, why the first term obtained in the RHS is actually $ \left[\frac{d\vec{u}}{dt}\right]_r $
That is the question, now I explain myself.
I have read a few articles where this equation is derived, Wikipedia for example. It starts by defining the unit vectors in the rotating frame $\hat{i} = (\textrm{cos}(w t), \textrm{sin}(w t), 0)$ and $\hat{j} = (-\textrm{sin}(w t), \textrm{cos}(w t), 0)$ where $w = ||\vec{w}||$ is the magnitude of the angular velocity of the rotating frame, assuming the rotation is performed around the $z$ axis. It is clear that this description of $\hat{i}$ and $\hat{j}$ is made from the fixed axis point of view (from the rotating axis, it would be $\left[\hat{i}\right]_r = (1,0,0)$ and $\left[\hat{j}\right]_r = (0,1,0)$, right?)
Then, the differentiation is made:
$$ \left[\frac{d\vec{u}}{dt}\right]_f = \frac{d (u_x \hat{i} + u_y \hat{j} + u_z \hat{k})}{dt} $$
In this last expression, it seems to me (and perhaps I am wrong here) that $u_x, u_y, u_z$ are the coordinates of the vector $\vec{u}$ as seen from the rotating frame. In other words: $\left[\vec{u}\right]_r = (u_x, u_y, u_z)$. On the other hand $\left[\vec{u}\right]_f = u_x \hat{i} + u_y \hat{j} + u_z \hat{k}$. Of course $\left[\vec{u}\right]_r \neq \left[\vec{u}\right]_f$ except at the times when both frames are aligned. I have a feeling that my confusion has something to do with what I've written in this paragraph.
I will omit the next steps in the calculations, the result is:
$$ \left[\frac{d\vec{u}}{dt}\right]_f = \left(\frac{du_x}{dt} \hat{i} + \frac{du_y}{dt} \hat{j} + \frac{du_z}{dt} \hat{k}\right) + \vec{w} \times \vec{u}$$
It is then stated that the term between parentheses on the RHS is $\left[\frac{d\vec{u}}{dt}\right]_r$. But this confuses me, I would have said that $\left[\frac{d\vec{u}}{dt}\right]_r = \left(\frac{du_x}{dt} , \frac{du_y}{dt}, \frac{du_z}{dt}\right)$ and again, by multiplying each component by its corresponding basis vector, I get the corresponding vector in the fixed frame, i.e. $ \left[\frac{d\vec{u}}{dt}\right]_f = \frac{du_x}{dt} \hat{i} + \frac{du_y}{dt} \hat{j} + \frac{du_z}{dt} \hat{k}$
I would really appreciate if someone could point out where is it that my confusion arises.