Complex analysis question...

In...

Cauchy-Riemann Equations -- from Wolfram MathWorld

... is written that, given the function of x and y...

$\displaystyle f(x,y) = u(x,y) + i\ v(x,y)$ (1)

... and setting...

$\displaystyle z= x + i\ y \implies dz= dx + i\ dy $ (2)

... is...

$\displaystyle \displaystyle \frac{df}{dz} = \frac{\partial {f}}{\partial {x}}\ \frac{\partial {x}}{\partial {z}} + \frac{\partial {f}}{\partial {y}}\ \frac{\partial {y}}{\partial {x}} = \frac{1}{2}\ (\frac{\partial {f}}{\partial {x}} - i\ \frac{\partial {f}}{\partial {y}}) $ (3)

It would for me very interesting to have more detail about the steps to arrive at the result (3). Any help will be greatly appreciated!...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$