the rules of complex differentiation
Happily, all the familiar rules of differentiation from real calculus carry over word for word to the complex derivative. If you can differentiate in calculus class, you can differentiate holomorphic functions — the algebra is identical, only the symbols now stand for complex numbers.
The rules are: linearity, (a f + b g)' = a f' + b g' for complex constants a, b; the product rule, (f g)' = f' g + f g'; the quotient rule, (f/g)' = (f' g - f g')/g^2 at points where g is nonzero; and the chain rule, (f(g(z)))' = f'(g(z)) g'(z). Each is proved exactly as in the real case, because the proofs only use limits of difference quotients and the arithmetic of the field — and the complex numbers are a field with limits, just like the reals.
Two consequences worth flagging: the rules guarantee that any function built from z by the four arithmetic operations and composition (a rational function, say) is automatically holomorphic wherever no denominator vanishes — you never need to recheck Cauchy-Riemann by hand. And because the same closure holds for power series, the rules are what let term-by-term differentiation work, eventually proving f' is itself holomorphic.
Differentiate f(z) = (z^2 + 1)/(z - i) by the quotient rule: f'(z) = [2z(z - i) - (z^2 + 1)(1)]/(z - i)^2 = (z^2 - 2 i z - 1)/(z - i)^2, valid for all z not equal to i.
Same rules as real calculus — the only change is that constants and values may be complex.
These rules apply to holomorphic functions only; they say nothing about z-bar, |z|, or Re z, which are not holomorphic and must not be 'differentiated' by these formulas.