The limit definition of a derivative is as rigorous as it is tedious. Thankfully there are a handful of rules which act as short-hand for the difference quotient’s evaluation. Once we learn these, the limit definition becomes an antiquated theorem seemingly used only for exam questions. Rather than a line, the complex numbers form a plane with one variable. Instead of left and right, a limit there must account for the infinitely many paths one could take to the point in question. We will explore this restriction and show how it leads to the Cauchy-Reimann Equations and how a power series follows from a complex derivative.
