The question

Let z = x + iy, with x and y real

If $\displaystyle f(z) = x^3 + iy^3$, find the value of the derivative of f at every point z where the derivative exists. Where is f analytic?

My attempt:

I used Cauchy Riemann sums as follows,

$\displaystyle u_x = 3x^2$

$\displaystyle u_y = 0$

$\displaystyle v_x = 0$

$\displaystyle v_y = 3y^2$

The equations hold when $\displaystyle 3x^2 = 3y^2$. This equals |x| = |y| which are intersecting lines through the origin of the x-y plane. So, the function has a derivative on these intersecting lines. If my understanding of analyticity is correct, this function is analytic nowhere since there's no epsilon neighbourhood where the function is differentiable.

My issue is, how do I find the value of the derivative? Do I use the definition of a derivative, or is there an easy way to find it using my result above?

Thanks.