Let $\displaystyle \vec A$ be a constant vector and let $\displaystyle f(\vec r)=\vec A \cdot \vec r$. Calculate the gradient of $\displaystyle f$ in Cartesian, cylindrical and spherical coordinates.

My attempt: $\displaystyle \vec \nabla \cdot f(\vec r)= \underbrace{(\vec \nabla \cdot \vec A)}_{=0 \text{ since A is constant}}\cdot \vec r+(\vec \nabla \cdot \vec r)\cdot \vec A$. I don't know if I'm right.
In Cartesian coordinates it gives $\displaystyle \frac{a_1 \partial r_1}{\partial x}\hat i+\frac{a_2 \partial r_2}{\partial y}\hat j+\frac{a_3 \partial r_3}{\partial z}\hat k$. That was assuming $\displaystyle \vec A=(a_1,a_2,a_3) $ and $\displaystyle \vec r =(r_1,r_2,r_3)$.

Now in cylindrical coordinates, can I assume $\displaystyle \vec A=(a_1, \varphi _1, z_1)$ and $\displaystyle \vec r=(r,\varphi, z)$? Anyway I'm totally stuck.