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 ux=f'(r(x,y,z) times rx is used correctly. But the second partial is:
, using the product rule.
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.