What does f '(z) means? 
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I see the derivative of 'w' with respect to 'z' all the time, what does it mean though? Derivative of 'w' with respect to 'theta' makes sense which is a vector on the argand graph.
 A: Differentiable complex functions are locally just like rotation and rescaling. Rotation by arg $f'(z)$ and rescaling by $|f(z)|$.
 I will give you some intuition about why that is the case, but before that, a definition.
A function $f$ is holomorphic at $z$ if the limit
$$ f'(z)= \lim_{h\rightarrow0} \frac{f(z-h)}{h}$$ exists and is finite.
A function $f$ is holomorphic on a subset of the complex plane if for all $z$ in that subset $f'(z)$ is a well defined continuous function.
Now you know from high-school that if $f$ is a real, differentiable function at $x_0$ you can write a linear approximation of $f$ around $x_0$. That is, for $x-x_0$ small enough
$$f(x) \sim f(x_0) + f'(x_0) (x-x_0) $$       
Now, for pretty much the same reasons, if $f$ is a complex function,  holomorphic at $z_0$, then for |z-z_0| small enough 
$$f(z) \sim f(z_0) + f'(z_0) (z-z_0) $$
Hopefully, now you can see that, while the derivative of a real function can be thought of as the slope of the graph, one can think of the derivative of a complex function as the rotation and stretching of the complex plane by the function. Can you see why?
There is one caveat: if $f'(z)=0$, then this stretching and rotating picture breaks down. This points are called critical points. Your function has one at 0.
A: Well, according to the image, $w$ is $f(z)$. Its derivative with respect to $z$ is...
Generally, $w$ is used to represent 'another' complex plane. The complex $z$-plane is generally the domain or original complex plane, and the complex $w$-plane is generally the codomain or transformed complex plane. There isn't really much to it, it's just a 'convention' letter like $y=f(x)$ for real functions.
