List of derivatives and integrals in alternative calculi

There are many alternatives to the classical calculus of Newton and Leibniz; for example, each of the infinitely many non-Newtonian calculi. ^[1] Occasionally an alternative calculus is more suited than the classical calculus for expressing a given scientific or mathematical idea. ^{[2] [3] [4]}

The table below is intended to assist people working with the alternative calculus called the "geometric calculus" (or its discrete analog). Interested readers are encouraged to improve the table by inserting citations for verification, and by inserting more functions and more calculi.

Table

In the following table;

${\displaystyle \psi (x)={\frac {\Gamma '(x)}{\Gamma (x)}}}$ is the digamma function,

${\displaystyle \operatorname {K} (x)=e^{\zeta ^{\prime }(-1,x)-\zeta ^{\prime }(-1)}=e^{{\frac {z-z^{2}}{2}}+{\frac {z}{2}}\ln(2\pi )-\psi ^{(-2)}(z)}}$ is the K-function,

${\displaystyle (!x)={\frac {\Gamma (x+1,-1)}{e}}}$ is subfactorial,

${\displaystyle B_{a}(x)=-a\zeta (-a+1,x)}$ are the generalized to real numbers Bernoulli polynomials.

Function ${\displaystyle f(x)}$	Derivative ${\displaystyle f'(x)}$	Integral ${\displaystyle \int f(x)dx}$ (constant term is omitted)	Multiplicative derivative ${\displaystyle f^{*}(x)}$	Multiplicative integral ${\displaystyle \int f(x)^{dx}}$ (constant factor is omitted)	Discrete derivative (difference) ${\displaystyle \Delta f(x)}$	Discrete integral (antidifference) ${\displaystyle \Delta ^{-1}f(x)}$ (constant term is omitted)	Discrete multiplicative derivative [5] (multiplicative difference)	Discrete multiplicative integral [6] (indefinite product) ${\displaystyle \prod _{x}f(x)}$ (constant factor is omitted)
${\displaystyle a}$	${\displaystyle 0}$	${\displaystyle ax}$	${\displaystyle 1}$	${\displaystyle a^{x}}$	${\displaystyle 0}$	${\displaystyle ax}$	${\displaystyle 1}$	${\displaystyle a^{x}}$
${\displaystyle x}$	${\displaystyle 1}$	${\displaystyle {\frac {x^{2}}{2}}}$	${\displaystyle {\sqrt[{x}]{e}}}$	${\displaystyle {\frac {x^{x}}{e^{x}}}}$	${\displaystyle 1}$	${\displaystyle {\frac {x^{2}}{2}}-{\frac {x}{2}}}$	${\displaystyle 1+{\frac {1}{x}}}$	${\displaystyle \Gamma (x)}$
${\displaystyle ax+b}$	${\displaystyle a}$	${\displaystyle {\frac {ax^{2}+2bx}{2}}}$	${\displaystyle \exp \left({\frac {a}{ax+b}}\right)}$	${\displaystyle {\frac {(b+ax)^{{\frac {b}{a}}+x}}{e^{x}}}}$	${\displaystyle a}$	${\displaystyle {\frac {ax^{2}+2bx-ax}{2}}}$	${\displaystyle 1+{\frac {a}{ax+b}}}$	${\displaystyle {\frac {a^{x}\Gamma ({\frac {ax+b}{a}})}{\Gamma ({\frac {a+b}{a}})}}}$
${\displaystyle {\frac {1}{x}}}$	${\displaystyle -{\frac {1}{x^{2}}}}$	${\displaystyle \ln |x|}$	${\displaystyle {\frac {1}{\sqrt[{x}]{e}}}}$	${\displaystyle {\frac {e^{x}}{x^{x}}}}$	${\displaystyle -{\frac {1}{x+x^{2}}}}$	${\displaystyle \psi (x)}$	${\displaystyle {\frac {x}{x+1}}}$	${\displaystyle {\frac {1}{\Gamma (x)}}}$
${\displaystyle x^{a}}$	${\displaystyle ax^{a-1}}$	${\displaystyle {\frac {x^{a+1}}{a+1}}}$	${\displaystyle e^{\frac {a}{x}}}$	${\displaystyle e^{-ax}x^{ax}}$	${\displaystyle (x+1)^{a}-x^{a}}$	${\displaystyle a\notin \mathbb {Z} ^{-}\,;}$${\displaystyle {\frac {B_{a+1}(x)}{a+1}},}$ ${\displaystyle a\in \mathbb {Z} ^{-}\,;}$${\displaystyle {\frac {(-1)^{a-1}\psi ^{(-a-1)}(x)}{\Gamma (-a)}},}$	${\displaystyle \left(1+{\frac {1}{x}}\right)^{a}}$	${\displaystyle \Gamma (x)^{a}}$
${\displaystyle a^{x}}$	${\displaystyle a^{x}\ln a}$	${\displaystyle {\frac {a^{x}}{\ln a}}}$	${\displaystyle a}$	${\displaystyle a^{\frac {x^{2}}{2}}}$	${\displaystyle (a-1)a^{x}}$	${\displaystyle {\frac {a^{x}}{a-1}}}$	${\displaystyle a}$	${\displaystyle a^{\frac {x^{2}+x}{2}}}$
${\displaystyle {\sqrt[{x}]{a}}}$	${\displaystyle -{\frac {{\sqrt[{x}]{a}}\ln a}{x^{2}}}}$	${\displaystyle x{\sqrt[{x}]{a}}-\operatorname {Ei} \left({\frac {\ln a}{x}}\right)\ln a}$	${\displaystyle a^{-{\frac {1}{x^{2}}}}}$	${\displaystyle a^{\ln x}}$	${\displaystyle a^{\frac {1}{1+x}}-a^{\frac {1}{x}}}$	${\displaystyle$ ?}	${\displaystyle a^{-{\frac {1}{x+x^{2}}}}}$	${\displaystyle a^{\psi (x)}}$
${\displaystyle \log _{a}x}$	${\displaystyle {\frac {1}{x\ln a}}}$	${\displaystyle \log _{a}x^{x}-{\frac {x}{\ln a}}}$	${\displaystyle \exp \left({\frac {1}{x\ln x}}\right)}$	${\displaystyle {\frac {(\log _{a}x)^{x}}{e^{\operatorname {li} (x)}}}}$	${\displaystyle \log _{a}\left({\frac {1}{x}}-1\right)}$	${\displaystyle \log _{a}\Gamma (x)}$	${\displaystyle \log _{x}(x+1)}$	${\displaystyle$ ?}
${\displaystyle x^{x}}$	${\displaystyle x^{x}(1+\ln x)}$	${\displaystyle$ ?}	${\displaystyle ex}$	${\displaystyle e^{-{\frac {1}{4}}x^{2}(1-2\ln x)}}$	${\displaystyle (x+1)^{x+1}-x^{x}}$	${\displaystyle$ ?}	${\displaystyle {\frac {(x+1)^{x+1}}{x^{x}}}}$	${\displaystyle \operatorname {K} (x)}$
${\displaystyle \Gamma (x)}$	${\displaystyle \Gamma (x)\psi (x)}$	${\displaystyle$ ?}	${\displaystyle e^{\psi (x)}}$	${\displaystyle e^{\psi ^{(-2)}(x)}}$	${\displaystyle (x-1)\Gamma (x)}$	${\displaystyle (-1)^{x+1}\Gamma (x)(!(-x))}$	${\displaystyle x}$	${\displaystyle {\frac {\Gamma (x)^{x-1}}{\operatorname {K} (x)}}}$
${\displaystyle \sin(ax)}$	${\displaystyle a\cos(ax)}$	${\displaystyle -{\dfrac {\cos(ax)}{a}}}$	${\displaystyle e^{a\cot(ax)}}$	${\displaystyle$ ?}	${\displaystyle \sin(a(x+1))-\sin(ax)}$	${\displaystyle -{\dfrac {1}{2}}\csc \left({\dfrac {a}{2}}\right)\cos \left({\dfrac {a}{2}}-ax\right)}$	${\displaystyle \cos(a)+\sin(a)\cot(ax)}$	${\displaystyle$ ?}

See also

• Derivative
• Differentiation rules
• Indefinite product
• Product integral
• Fractal derivative

References

1. Grossman, Michael; Katz, Robert (1972). Non-Newtonian calculus. Pigeon Cove, Mass.: Lee Press. ISBN   0-912938-01-3. OCLC   308822.
2. Bashirov, Agamirza E.; Kurpınar, Emine Mısırlı; Özyapıcı, Ali (1 January 2008). "Multiplicative calculus and its applications". Journal of Mathematical Analysis and Applications . 337 (1): 36–48. Bibcode:2008JMAA..337...36B. doi: 10.1016/j.jmaa.2007.03.081 .
3. Filip, Diana Andrada; Piatecki, Cyrille (2014). "A non-Newtonian examination of the theory of exogenous economic growth". Mathematica Eterna . 4 (2): 101–117.
4. Florack, Luc; van Assen, Hans (January 2012). "Multiplicative Calculus in Biomedical Image Analysis". Journal of Mathematical Imaging and Vision . 42 (1): 64–75. doi: 10.1007/s10851-011-0275-1 . ISSN   0924-9907. S2CID   254652400 – via SpringerLink.
5. Khatami, Hamid Reza; Jahanshahi, M.; Aliev, N. (5–10 July 2004). An analytical method for some nonlinear difference equations by discrete multiplicative differentiation (PDF). Dynamical Systems and Applications, Proceedings. Antalya, Turkey. pp. 455–462. Archived from the original (PDF) on 6 Jul 2011.
6. Jahanshahi, M.; Aliev, N.; Khatami, Hamid Reza (5–10 July 2004). An analytic-numerical method for solving difference equations with variable coefficients by discrete multiplicative integration (PDF). Dynamical Systems and Applications, Proceedings. Antalya, Turkey. pp. 425–435. Archived from the original (PDF) on 6 Jul 2011.

External links

• Non-Newtonian calculus website