Let $f:\mathbb C\to\mathbb C$, and write it as follows: \(f(z) = u(x,y) + iv(x,y)\) where $z\in \mathbb C$ is such that $z = x+iy$ and $u,v:\mathbb R^2\to \mathbb R$. Then $f$ is complex differentiable at $z_0\in \mathbb C$ with derivative $f’(z_0)$ if the limit \(f'(z_0) = \lim_{z\to z_0} \frac{f(z)-f(z_0)}{z-z_0}\) exists.