Definition 1. Operator $\nabla$ is defined as \begin{equation} \nabla = \mathbf{i} \partial_x + \mathbf{j} \partial_y+ \mathbf{k} \partial_z. \label{equ-Z.1} \end{equation} It could be applied to a scalar function resulting in its gradient ($\operatorname{grad}\phi$) \begin{equation*} \nabla \phi = \mathbf{i} \partial_x\phi + \mathbf{j} \partial_y\phi+ \mathbf{k} \partial_z\phi \end{equation*} and to vector function $\mathbf{A}=A_x\mathbf{i}+A_y\mathbf{j}+A_z\mathbf{k}$ resulting in its divergence ($\operatorname{div}\mathbf{A}$) \begin{equation*} \nabla \cdot \mathbf{A} = \partial_xA_x + \partial_y A_y+ \partial_zA_z \end{equation*} and also in its curl ($\operatorname{curl}\mathbf{A}$) or rotor ($\operatorname{rot}\mathbf{A}$), depending on the mathematical tradition: \begin{equation*} \nabla \times \mathbf{A} = \left| \begin{matrix} \mathbf{i} & \mathbf{j} &\mathbf{k} \\ \partial_x & \partial_y & \partial_z\\ A_x & A_y &A_z\end{matrix}\right|. \end{equation*}
Definition 2. \begin{equation} \Delta= \nabla^2 = \nabla\cdot \nabla= \partial^2_x + \partial^2_y+ \partial^2_z. \label{equ-Z.2} \end{equation} is Laplace operator or simply Laplacian.
Four formulae to remember: \begin{gather} \nabla (\nabla \phi)= \Delta \phi,\label{equ-Z.3}\\[3pt] \nabla \times (\nabla \phi)= 0,\label{equ-Z.4}\\[3pt] \nabla \cdot (\nabla \times \mathbf{A})= 0,\label{equ-Z.5}\\[3pt] \nabla \times (\nabla \times \mathbf{A})= -\Delta \mathbf{A} + \nabla (\nabla \cdot \mathbf{A})\label{equ-Z.6} \end{gather} where all but the last one are obvious and the last one follows from \begin{equation} \mathbf{a}\times (\mathbf{a} \times \mathbf{b})= - \mathbf{a}^2 \mathbf{b}+ (\mathbf{a}\cdot \mathbf{b}) \mathbf{a} \label{equ-Z.7} \end{equation} which is the special case of \begin{equation} \mathbf{a}\times (\mathbf{b} \times \mathbf{c})= \mathbf{b}(\mathbf{a}\cdot\mathbf{c})- \mathbf{c}(\mathbf{a}\cdot\mathbf{b}). \label{equ-Z.8} \end{equation}
Recall Leibniz rule how to apply the first derivative to the product which can be symbolically written as \begin{equation*} \partial (uv)= (\partial_u + \partial_v)(uv)= \partial_u (uv)+\partial_v (uv)= v\partial_u (u) +u\partial_v (v)=v\partial u +u\partial v \end{equation*} where subscript "$u$" means that it should be applied to $u$ only.
Since $\nabla$ is a linear combination of the irst derivatives, it inherits the same rule. Three formulae are easy \begin{gather} \nabla ( \phi\psi)= \phi\nabla \psi +\psi \nabla \phi,\label{equ-Z.9}\\[3pt] \nabla \cdot ( \phi\mathbf{A})= \phi\nabla \cdot \mathbf{A} +\nabla \phi\cdot \mathbf{A} ,\label{equ-Z.10}\\[3pt] \nabla \times ( \phi\mathbf{A})= \phi\nabla \times \mathbf{A} +\nabla \phi\times \mathbf{A} ,\label{equ-Z.11}\end{gather} and the fourth follows from the Leibniz rule and (\ref{equ-Z.8}) \begin{align} \nabla \times ( \mathbf{A}\times \mathbf{B})= &(\mathbf{B}\cdot\nabla)A-\mathbf{B}(\nabla\cdot \mathbf{A})\notag\\ - &(\mathbf{A}\cdot\nabla)B-\mathbf{A}(\nabla\cdot \mathbf{B}).\label{equ-Z.12} \end{align}