why Laplacian in Cartesian and cylindrical coordinates produces different results?

Question

I tried to calculate Laplacian for function $f = \frac{k}{r}$ in both Cartesian and cylindrical coordinates, but i got different results. To calculate Laplacian in Cartesian coordinates, first i converted function $f = \frac{k}{r}$ to rectangular form $f = \frac{k}{\sqrt{x^2+y^2+z^2}}$, then used $\nabla^2f = \frac{\partial^2f}{\partial x^2}+\frac{\partial^2f}{\partial y^2}+\frac{\partial^2f}{\partial z^2}$ formula and i got zero. To calculate Laplacian in cylindrical coordinates, i used $\nabla^2f = \frac{1}{r}\frac{\partial}{\partial r}(r\frac{\partial f}{\partial r})+\frac{1}{r^2}\frac{\partial^2 f}{\partial \phi^2}+\frac{\partial^2 f}{\partial z^2}$ formula and i got $\frac{k}{r^3}$. I am sure something in my calculations is not rigth, but i don't know what it is. Why i got two different result?