$f(z)=u+iv$ * Given,$f(z)=x^2-y^2+2ixy$ $\therefore u=x^2-y^2$ ...(1) $v=2xy$ ...(2) * Differentiating equation (1) w.r.t 'x', $ u_x=2x$ * Again, differentiating equation (1) w.r.t 'y', $ u_y=-2y $ * Now, differentiating equation (2) w.r.t 'x', $ v_x=2y$ * Again differentiating equation (2) w.r.t 'y', $v_y=2x$ $\therefore u_x=v_y$ * Also,$u_y=-v_x$ * This satisfies the Cauchy-Riemann equations. * $\therefore f(z)$ is analytic. * Now, by Milne-Thompson method, put $x=z,y=0$ $\therefore f(z)=z^2$ * Differentiating both sides w.r.t 'z', $f'(z)=2z$ * We know,$z=x+iy$ $\therefore f'(z)=2(x+iy)$ $\therefore f'(z)=2x+2iy$