Yes, you made an error in the calculation of the 2nd partials. Here's the situation: you have a function r(x,y,z) and a function u=f(r) (in your case 1/r). So you want the partials of u w.r.t. x , y and z. The first partial u_{x}=f'(r(x,y,z)timesr_{x}is used correctly. But the second partial is:

, using theproductrule.

If you do this (replacing occurrences of r) you get:

Similarly for the other 2nd partials. It is then easy to see that Laplace's equation is satisfied.