Calculate the ux and uy

Question

Laplacian

Solve the following problem:

In the closed unit disc U={(x,y)|x²+y²≤ 1}, use r:=√x²+y² and consder the function u given by
        u(x,y)={xy In In e/r     if r≠0
                   {0                   if r=0

(1) Show u∈C¹(U), and calculate u_x, u_y.

(2) Calculate u_xx, u_yy and u_xy and show that u_xx and u_yy are continuous in the whole disk, but that u?c². Calculate f := - Δu.