Computing d/dz(z²)

Complex Analysis · Axiom Academy

Proving the power rule for z^2 straight from the complex-derivative limit Let f(z) = z^2 be a function of the complex variable z . Using the definition of the complex derivative, show that f'(z) = 2z — the same power rule we know from real calculus. Nicely done. You proved the power rule for z^2 the rigorous way — straight from the limit — and the algebra collapsed to 2z no matter how h approached 0 . The familiar rule survives: , exactly as in real calculus. Direction never mattered: after canceling h you were left with 2z + h , which tends to 2z as from any direction in the plane — real, imaginary, or diagonal. That is precisely what it means for f to be complex-differentiable here. Contrast with : for the same steps give , which is 1 along the real axis but -1 along the imaginary axis. The limit disagrees by direction, so is not differentiable — the cancellation that saved z^2 never happens. The formula predicts f'(1+i) = 2(1+i) = 2 + 2i . Two difference quotients with a tiny h confirm it from two different directions: Both land on 2 + 2i , just as expected. The very same binomial trick works for any positive integer n — every term past the first carries an extra factor of h and vanishes: The real lesson isn't the answer 2z — it's why it exists: differentiability in the complex plane demands the limit agree from every direction, and z^2 passes that test while fails it.

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