Find all $z = x+iy$ such that $f(x+iy) = (x^2 - y^2) + i(x^{3} + y^{3})$ is complex differentiable at $z$.
I began by computing the partial derivatives of $f$ \begin{align*} \frac{\partial f}{\partial x} &= 2x + i3x^2\\ \frac{\partial f}{\partial y} &= -2y + i3y^2 \end{align*}
They definitely exist everywhere and I think they are continuous everywhere. So we know from the Cauchy-Riemann equations that $f$ will be differentiable when $f_{x}$ = $-if_{y}$. Comparing the real parts of each and the imaginary parts of each this is when \begin{align*} 2x &= 3y^{2} \\ 3x^{2} &= 2y \end{align*} The solutions of this system are $(0,0)$ and $(\frac{2}{3},\frac{2}{3})$, so $f$ is complex differentiable at $z = 0$ and $z = \frac{2}{3} + i\frac{2}{3}$.
Is there anything wrong here?