When trying to use the del function on the following equation (1) representing a linearly polarized, monochromatic, plane waves traveling in the z-direction:
$$\textbf{E} (\textbf{r},t) = \textbf{E}_0 e^{i(k_z z- \omega t)} \tag{1}$$
where $k_z$ is the z-component of the vector $k$.
When trying to obtain $\nabla \textbf{E}$, and knowing that there is only a $z$-component on equation (1), I wrote the expression as:
$$\nabla \textbf{E} = \frac{\partial \textbf{E}}{\partial z} = \textbf{E}_0 e^{-i \omega t} \left[ik_z e^{ik_z z} \right] + ... \tag{2}$$
I am struggling to obtain the second half of this differentiation because $k$ is a z component and I don't know if that should be simply considered to be a constant or if I should also differentiate it. If the latter is correct, how must I differentiate it?