Let \(\Omega\subseteq\CC\) be open and let \(f = u+iv\colon\Omega\to\CC\), where \(u,v\colon\Omega\to\RR\) are continuously differentiable as functions of \((x,y)\), \(z=x+iy\). If \(u_x=v_y\) and \(u_y=-v_x\) on \(\Omega\), then \(f\) is holomorphic on \(\Omega\) and \[ f'(z) = \del f(z) = u_x(z) + iv_x(z). \]
Fix \(z\in\Omega\) and write \(h=h_1+ih_2\). Since \(u\) and \(v\) are continuously differentiable, they are real differentiable at \(z\), so \[ f(z+h)-f(z)=u_xh_1+u_yh_2+i\qty{v_xh_1+v_yh_2}+o(\abs{h}), \] with the partial derivatives evaluated at \(z\). Substituting \(u_y=-v_x\) and \(v_y=u_x\) gives \(u_xh_1-v_xh_2+i\qty{v_xh_1+u_xh_2}=(u_x+iv_x)(h_1+ih_2)\), so \(\frac{f(z+h)-f(z)}{h}\to u_x+iv_x\) as \(h\to0\). Finally, \(\del f=\frac12\qty{f_x-if_y}=\frac12\qty{u_x+v_y+i(v_x-u_y)}=u_x+iv_x\) by the same equations.