the derivative of z^n and polynomials

The single most-used derivative in the subject is the power rule for z to a whole number: the derivative of z^n is n z^(n-1), the same compact formula you know from real calculus. From it, every polynomial differentiates termwise, so the engine of computation in complex analysis starts here.

You can prove it directly from the difference quotient without memorizing anything. For n a positive integer, (z^n - z_0^n)/(z - z_0) factors algebraically as z^(n-1) + z^(n-2) z_0 + ... + z_0^(n-1), a sum of n terms; as z -> z_0 each term tends to z_0^(n-1), giving n z_0^(n-1). The rule extends to negative integer powers (so the derivative of 1/z = z^(-1) is -z^(-2), valid off z = 0) and, once branches are fixed, to general complex exponents.

Combined with linearity, this instantly differentiates any polynomial: the derivative of a_n z^n + ... + a_1 z + a_0 is n a_n z^(n-1) + ... + a_1. Because polynomials and (off their poles) rational functions are therefore holomorphic with easily computed derivatives, they are the first large supply of worked examples — and the building blocks for Taylor series, where each coefficient is a scaled higher derivative at a point.