Rank of the differential of a holomorphic function Given a holomorphic function $\newcommand{\d}{\mathrm{d}}$
$$f: U \subset \mathbb{C} \to \mathbb{C},$$
 I want to show that the rank of $\d f$ at any point $x \in U$ cannot be $1$ (i.e. must be $0$ or $2$). 
I'm unsure on what to do. I thought to express it in terms of $f=u(x,y) + iv(x,y)$ and then $\d f$ will be the Jacobian matrix. Assuming the first row is a multiple of the second, I get that 
$$\partial_x u = -a \, \partial_y u$$
$$\partial_x v = a \, \partial_y v$$
for some $a \in \mathbb{C}$, where we must have $a = \pm i$ from the Cauchy Riemann equations.
This doesn't seem useful though. Help?

Hint: Show that $Df(a)((x,y)) = f'(a)(x+iy).$ Here the real world meets the complex world: The left side is the real differential of $f$ at $a,$ viewed as acting on $(x,y)\in \mathbb R^2.$ The right side is complex mulitplication of the complex derivative $f'(a)$ with the complex number $x+iy.$
