As always with partial derivatives, you have to ask what the "other variable(s)" are. When you see the notation , it is implicitly telling you that x is a function of z and some other (unspecified) variable. The usual convention when dealing with functions of a complex variable is that the "other variable" is the complex conjugate , so that and . It's then clear that and , and equation (3) follows by using the chain rule. Notice that f is meant to be an analytic function, so it is not dependent on . Thus the derivative can be written with straight d's rather than as a partial derivative.
In books on complex function theory, the condition for f to be analytic is sometimes written In fact, the condition is exactly equivalent to the Cauchy–Riemann equations.