Let \(\Omega\subseteq\CC\) be open, let \(f=u+iv\colon\Omega\to\CC\) with \(u,v\) real-valued, and regard \(f\), \(u\), \(v\) as functions of \((x,y)\) where \(z=x+iy\). If \(f\) is complex differentiable at \(z_0=x_0+iy_0\in\Omega\), then the partial derivatives of \(u\) and \(v\) exist at \((x_0,y_0)\), \[ f'(z_0)=\dd{f}{x}(x_0,y_0)={1\over i}\dd{f}{y}(x_0,y_0), \] and at \((x_0,y_0)\) \[ \dd{u}{x} = \dd{v}{y}, \qquad \dd{u}{y} = -\dd{v}{x}. \]
The limit \(f'(z_0)=\lim_{h\to0}\frac{f(z_0+h)-f(z_0)}{h}\) exists, so it may be computed along any direction. Along real \(h=h_1\in\RR\), \[ f'(z_0) = \lim_{h_1\to 0} { f(x_0+h_1, y_0) - f(x_0, y_0) \over h_1} = \dd{f}{x}(x_0, y_0). \] Along purely imaginary \(h=ih_2\) with \(h_2\in\RR\), \[ f'(z_0) = \lim_{h_2\to 0} { f(x_0, y_0+h_2) - f(x_0, y_0) \over ih_2 } = {1\over i} \dd{f}{y}(x_0, y_0). \] Taking real and imaginary parts of these limits shows that the partial derivatives of \(u\) and \(v\) exist. Writing \(1/i=-i\), \[ \begin{aligned} \dd{f}{x} &= \dd{u}{x} + i \dd{v}{x}, \\ {1\over i} \dd{f}{y} &= -i \qty{ \dd{u}{y} + i \dd{v}{y}} = \dd{v}{y} - i\dd{u}{y}. \end{aligned} \] Equating real and imaginary parts of the two expressions for \(f'(z_0)\) gives \(\dd{u}{x} = \dd{v}{y}\) and \(\dd{v}{x} = -\dd{u}{y}\).