What Lies Between a Function and Its Derivative? | Fractional Calculus

Can you take a derivative only partway? Is there any meaning to a "half-derivative"? Does such a concept even make sense? And if so, what do these fractional derivatives look like? Previous video about Cauchy's Formula for Repeated Integration: https://youtu.be/jNpKKDekS6k A really nice video that derives the gamma function from scratch: https://youtu.be/v_HeaeUUOnc =Chapters= 0:00 - Interpolating between polynomials 1:16 - What should half derivatives mean? 3:56 - Deriving fractional integrals 8:22 - Playing with fractional integrals 9:12 - Deriving fractional derivatives 13:53 - Fractional derivatives in action 16:19 - Nonlocality 17:54 - Interpreting fractional derivatives 18:51 - Visualizing fractional integrals 22:10 - My thoughts on fractional calculus 23:10 - Derivative zoo =============================== MAIN SOURCES USED FOR THIS VIDEO Podlubny, Igor. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press, 1999 Podlubny, I.: "Geometric and physical interpretation of fractional integration and fractional differentiation." Fractional Calculus and Applied Analysis, vol. 5, no. 4, 2002, pp. 367--386. - (for the visualization trick for fractional integrals) Edmundo Capelas de Oliveira, José António Tenreiro Machado, "A Review of Definitions for Fractional Derivatives and Integral", Mathematical Problems in Engineering, vol. 2014, Article ID 238459, 6 pages, 2014. https://doi.org/10.1155/2014/238459 - (for the zoo of alternative fractional derivatives) =============================== Minor correction: The footnote at 7:34 should say the trig substitution produces another *whole* factor of pi (not a root pi) in the numerator which then cancels the *two* root(pi)'s that appear in the denominator from applying the half integral formula twice. =============================== CREDITS This video uses the song "Rubix Cube" coming courtesy of Audionautix.com =============================== Want to support future videos? Become a patron at https://www.patreon.com/morphocular Thank you for your support! =============================== The animations in this video were mostly made with a homemade Python library called "Morpho". If you want to play with it, you can find it here: https://github.com/morpho-matters/morpholib =============================== This video is part of the 3Blue1Brown Summer of Math Exposition 2 (#SoME2). You can find out more about it here: https://summerofmathexposition.substack.com/p/the-summer-of-math-exposition-is