How to show the transition from equation $(4)$ to equation $(5)$? How to show the transition between $(4)$ to $(5)$ in here?
 A: Equation $4$ is just an application of the multivariable chain rule, while equation $5$ uses the definition of $\dfrac{d}{dz}$.
Don't think of equation $5$ as having been derived from equation $4$. Instead, think of the right hand sides of equations $4$ and $5$ as two different expressions for the same quantity, namely $\dfrac{df}{dz}$.
A: $\frac{\partial}{\partial z} = \frac{1}{2}\left(\frac{\partial }{\partial x} - i \frac{\partial }{\partial y} \right)$ is the Wirtinger derivative and therefore
$$\frac{\partial x}{\partial z} = \frac{1}{2}, \quad \frac{\partial y}{\partial z} = -\frac{i}{2}.$$ 
Edit: Here are the intermediate steps (requested in a comment):
$$
\frac{\partial x}{\partial z} = \frac{1}{2}\left(\frac{\partial x }{\partial x} - i \frac{\partial x}{\partial y} \right) = \frac{1}{2} \left(1 -i\cdot0 \right)
= +\frac{1}{2} 
$$
$$
\frac{\partial y}{\partial z} = \frac{1}{2}\left(\frac{\partial y }{\partial x} - i \frac{\partial y}{\partial y} \right) = \frac{1}{2} \left(0 -i\cdot 1 \right) = -\frac{i}{2}
$$
