Do You REALLY Understand Derivative? | Symmetric Derivative and Generalized Pseudoderivative

You think you know the derivative? Think again. The classic definition, the limit of the forward difference quotient (or the backward), is only one way to define the slope of a tangent line. But what if we change the rules?

In this video, we dive deep into the fascinating world of generalized difference quotients and pseudo-derivatives. We explore non-standard schemes like the +2h and -h difference, and the infamous symmetric derivative, and ask a simple but profound question: Why did Cauchy and the other math giants settle on the definition we use today?


Links:
Proof of the Main Theorem
https://www.overleaf.com/read/wjgcbpf...

Wikipedia Page on Symmetric Derivative
https://en.wikipedia.org/wiki/Symmetr...

Chapters:
00:00 Intro (Definition of Derivative)
01:15 Discontinuity Obstructs Differentiability
04:00 Asymmetric Pseudoderivative is Almost Derivative
07:06 Symmetric Derivative Can Fail with Continuity
08:51 Usefulness of Symmetric Difference Quotient
10:02 Outro (Philosophy of Mathematical Definition)


Music🎵:
Between Whims - HOYO-MiX · 林一凡    • Between Whims  
Promenade of Tides - HOYO-MiX · 林一凡    • Promenade of Tides  
Snug Bundle of Warmth - Deer Suo · HOYO-MiX    • Snug Bundle of Warmth  
Driftveil City Funk Cover - Sean R. Hanson    • Pokemon BW - Driftveil City - Funk Cover