What does it mean to take a complex derivative? (visually explained)

The complex derivative, from differentials to the Cauchy-Riemann Equations
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The source code for the animations can be found here:
https://github.com/vivek3141/videos

A huge thanks to @3blue1brown , @Aleph0 , @alfcnz , Sumedh Shenoy, Nikhil Maserang and Oliver Ni for helping me review the video!

If you're interested in learning why conformal doesn't necessarily mean differentiable, read Chapter 4.VI "Conformal = Analytic" of Tristan Needham's "Visual Complex Analysis", which you can find here: http://usf.usfca.edu/vca/

This book is incredible, some of the ideas in this video were taken from this book. You can find all my sources here: https://docs.google.com/document/d/13...

Some extra info:
At 20:47, I mention that a function is holomorphic if it satisfies the cauchy-riemann equations. This is not always true, the partial derivatives have to be continuous as well.
For example, f(z) = {0 if z=0, z^5/|z^4| if z!=0} satisfies the cauchy-riemann equations but is not differentiable at z=0.
Thanks to Ge for pointing this out!

Mistakes:
3:15: The upper number line should still be labeled as "x" instead of "x^2"
14:56 It should be nz^(n-1)

These animation in this video was made using 3blue1brown's library, manim:
https://github.com/3b1b/manim

Chapters:
0:00 Intro
1:53 The Real Derivative, Revisited
4:31 Differential View
9:17 Transformation View
13:02 Conformality
16:01 Cauchy-Riemann Equations
22:15 Brilliant Ad, Stereographic Projection
23:57 Outro, deriv of e^z


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What does it mean to take a complex derivative? (visually explained)

Some tags: complex analysis, cauchy-riemann, imaginary, derivative, vcubingx