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).

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.