On Stein's and Shakarachi's Complex Analysis book, a proof is given to the following theorem:
$\textbf{Theorem 2.4}\,\,$ Suppose $f=u+i\,v\,$ is a complex-valued function defined on an open set $\,\Omega\subset\mathbb{C}$. If $u$ and $v$ are continuously differentiable and satisfy the Cauchy-Riemann equations on $\Omega$, then $f$ is holomorphic on $\,\Omega\,$ and $f'(z)=\partial(f)/\partial(z)$.
The proof is as follows:
Write $$u(x+h_1,y+h_2)-u(x,y)=\frac{\partial u}{\partial x}h_1+\frac{\partial u}{\partial y}h_2+|h|\psi_1(h)$$ and
$$v(x+h_1,y+h_2)-v(x,y)=\frac{\partial v}{\partial x}h_1+\frac{\partial v}{\partial y}h_2+|h|\psi_2(h)$$
where $\psi_j\to0$ (for $j=1,2$) as $|h|$ tends to $0$, and $h=h_1+i\,h_2$. Using the Cauchy-Riemann equations we find that $$f(z+h)-f(z)=\left(\frac{\partial u}{\partial x}-i\frac{\partial u}{\partial y}\right)(h_1+i\,h_2)+|h|\psi(h).$$ where $\psi(h)=\psi_1(h)+\psi_2(h)\to0$ as $|h|\to0$. Therefore, $f$ is holomorphic and $$f'(z)=2\frac{\partial u}{\partial z}=\frac{\partial f}{\partial z}.\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\square$$
Now, my question is, where does the $$2\partial u /\partial z$$ come from?
I don't have access to the book right now, but my gess is that earlier the authors defines$$\frac{\partial u}{\partial z}=\frac12\left(\frac{\partial u}{\partial x}-i\frac{\partial u}{\partial y}\right).$$