Differentiation Rules

Complex Analysis · Axiom Academy

Why the familiar rules survive in ℂ — and what a complex derivative secretly demands. 1. A Derivative From Every Direction The complex derivative is defined by the same limit you already know — but z is now a point in the plane , free to shrink toward 0 along any path. For f'(z) to even exist , every one of those approaches must land on the same number . the same value, for every direction of approach Watch the probe arrow z swing through every direction while the difference quotient is read off. For an analytic f(z)=z^2 , the readout holds rock-steady at 2z — the arrows all collapse to one answer. That agreement is what "differentiable" means in ℂ. Only two directions to reconcile: from the left and from the right. Infinitely many directions in the plane — all must agree. A much higher bar. The conjugate f(z)= z looks harmless, yet it is differentiable nowhere . The reason is exactly the direction test from Step 1 — and it fails on the very first two directions you try. Approach along the real axis ( z = h ) and the quotient reads +1 . Approach along the imaginary axis ( z = ih ) and it reads −1 . Two directions, two different limits — so no single f'(z) can exist. The rules assume what they cannot supply So ( ^2 )' 2 : the power rule presupposes the derivative exists , and here it does not. The same trap catches (z) and |z| — always confirm differentiability before reaching for a rule. 3. Cauchy–Riemann: The Price of Agreement

This is the written version of the interactive lesson above. See the full Complex Analysis course.