the complex derivative

On the real line you learned that the derivative measures the slope of a curve — how fast the output changes as you nudge the input. The complex derivative asks the very same question, f'(z) = the rate of change of f at the point z, but now both the input z and the output f(z) are complex numbers, so 'nudging' z means moving it in any direction in the plane.

Formally f'(z_0) is the limit of the difference quotient (f(z) - f(z_0)) / (z - z_0) as z approaches z_0. The catch is that z can approach z_0 from infinitely many directions — along the real axis, the imaginary axis, a spiral, anything — and we demand the SAME limiting value no matter how it approaches. That one number, if it exists, is f'(z_0). For example, for f(z) = z^2 the quotient is (z^2 - z_0^2)/(z - z_0) = z + z_0, which tends to 2 z_0, so f'(z) = 2z, exactly as in the real case.

This direction-independence is the whole story of the subject. A real-differentiable map of the plane (thinking of z = x + i y) only needs its partial derivatives to exist and combine linearly; a complex-differentiable map must do something far more rigid, and that extra rigidity is encoded in the Cauchy-Riemann equations. The payoff is enormous: a function with even ONE complex derivative on an open set turns out to have infinitely many and to equal its own power series — a miracle with no real-variable analogue.