The ''[[Boolean derivative]]'' of the function to one of the arguments is a (''k''−1)-ary function that is true when the output of the function is sensitive to the chosen input variable; it is the XOR of the two corresponding cofactors. A derivative and a cofactor are used in a [[Reed–Muller expansion]]. The concept can be generalized as a ''k''-ary derivative in the direction dx, obtained as the difference (XOR) of the function at x and x + dx.<ref name=":1" />
The ''[[Zhegalkin polynomial#Möbius transformation|Möbius transform]]'' (or ''Boole-MöbiusBoole–Möbius transform'') of a Boolean function is the set of coefficients of its [[Zhegalkin polynomial|polynomial]] ([[algebraic normal form]]), as a function of the monomial exponent vectors. It is a [[Involution (mathematics)|self-inverse]] transform. It can be calculated efficiently using a [[Butterfly diagram|butterfly algorithm]] ("''Fast Möbius Transform''"), analogous to the [[Fast Fourier transform|Fast Fourier Transform]].<ref>{{Citation|last=Carlet|first=Claude|title=Boolean Functions for Cryptography and Error-Correcting Codes|date=2010|url=https://www.math.univ-paris13.fr/~carlet/chap-fcts-Bool-corr.pdf|work=Boolean Models and Methods in Mathematics, Computer Science, and Engineering|pages=257–397|editor-last=|editor-first=|series=Encyclopedia of Mathematics and its Applications|place=Cambridge|publisher=Cambridge University Press|isbn=978-0-521-84752-0|access-date=2021-05-17|editor2-last=|editor2-first=}}</ref> ''Coincident'' Boolean functions are equal to their Möbius transform, i.e. their truth table (minterm) values equal their algebraic (monomial) coefficients.<ref>{{Cite journal|last1=Pieprzyk|first1=Josef|last2=Wang|first2=Huaxiong|last3=Zhang|first3=Xian-Mo|date=2011-05-01|title=Mobius transforms, coincident Boolean functions and non-coincidence property of Boolean functions|url=https://doi.org/10.1080/00207160.2010.509428|journal=International Journal of Computer Mathematics|volume=88|issue=7|pages=1398–1416|doi=10.1080/00207160.2010.509428|s2cid=9580510 |issn=0020-7160}}</ref> There are 2^2^(''k''−1) coincident functions of ''k'' arguments.<ref>{{Cite journal|last1=Nitaj|first1=Abderrahmane|last2=Susilo|first2=Willy|last3=Tonien|first3=Joseph|date=2017-10-01|title=Dirichlet product for boolean functions|url=https://doi.org/10.1007/s12190-016-1037-4|journal=Journal of Applied Mathematics and Computing|language=en|volume=55|issue=1|pages=293–312|doi=10.1007/s12190-016-1037-4|s2cid=16760125 |issn=1865-2085}}</ref>
