So far we have supposed that $$y = \phi(x)$$ is a purely real function of $$x$$. If $$y$$ is a complex function $$\phi(x) + i\psi(x)$$, then we define the derivative of $$y$$ as being $$\phi'(x) + i\psi'(x)$$. The reader will have no difficulty in seeing that Theorems (1)–(5) above retain their validity when $$\phi(x)$$ is complex. Theorems (6) and (7) have also analogues for complex functions, but these depend upon the general notion of a ‘function of a complex variable’, a notion which we have encountered at present only in a few particular cases.