Transform Laplace's equation to polar coordinates.
Now what?
First read this: http://www.math.unc.edu/Faculty/pbe/...ations/HW8.pdf
I'll derive the first result (using your notation):
To get , differentiate implicitly with respect to x (use the chain rule on each side):
.
To get , differentiate implicitly with respect to x (use the quotient rule on the right hand side and the chain rule on the left hand side):
.
Therefore .
Now there's a bit of work left for you to do.