Understanding partial derivatives related to the Cauchy-Riemann equations So I am reading a soft introductory book on complex variables (I have complex analysis next year.) The book I am reading is called "Complex Variables Demystified". I am currently reading a chapter where the main goal is to arrive at the Cauchy-Riemann equations. I'm having problems following the deduction; specifically, given a function $f$ of a complex variable $z=x+iy$, the author starts out by expressing the partial derivatives of $f$ with respect to $z$ and $\bar z $. To do this, he first arrive at : 
$\frac{\partial}{\partial z}=\frac{\partial x}{\partial z}\frac{\partial}{\partial x}+\frac{\partial y}{\partial z}\frac{\partial}{\partial y}$, and similarly $\frac{\partial}{\partial \bar z}=\frac{\partial x}{\partial \bar  z}\frac{\partial}{\partial x}+\frac{\partial y}{\partial \bar z}\frac{\partial}{\partial y}$. 
At this point in the text i am a littlebit lost. First of all, how did he deduce these equations? (Where do they come from?) And what do they mean? Is the author expressing a general differential operator $\frac{\partial}{\partial z}$ of a complex variable $z=x+iy$ in which $x$ and $y$ again are funcions $x=x(z,\bar z)$, $y=y(z,\bar z)$ of $z$ and $\bar z$ ? And what does it mean in the two equations above when $\frac{\partial}{\partial x}$ is written without anything in the numerator? Is this again a general differential operator?
I'm essentially just trying to understand how the author gets to the final expressions here:
 
I would be happy if anyone could explain what's going on as simple as possible :) 
-Thanx 

is this correct? :) 
and this? Thanks for your response by the way!

