List of derivatives and integrals in alternative calculi

This is a table of derivatives and integrals in alternative calculi.

In the following table $\psi(x)=\frac{\Gamma'(x)}{\Gamma(x)}\,$ is the digamma function, $K(x)=e^{\zeta^\prime(-1,x)-\zeta^\prime(-1)}=e^{\frac{z-z^2}{2}+\frac z2 \ln (2\pi)-\psi^{(-2)}(z)}$ is the K-function, $(!x)=\frac{\Gamma(x+1,-1)}{e}$ is subfactorial, $B_a(x)=-a\zeta(-a+1,x)\,$ are the generalized to real numbers Bernoulli polynomials.

Function $f(x)\,$	Derivative $f'(x)\,$	Integral $\int f(x) dx$ (constant term is omitted)	Multiplicative derivative $f^*(x)\,$	Multiplicative integral $\int f(x)^{dx}$ (constant factor is omitted)	Discrete derivative (difference) $\Delta f(x)\,$	Discrete integral (antidifference) $\Delta^{-1} f(x)\,$ (constant term is omitted)	Discrete multiplicative integral[2] (indefinite product) $\prod _x f(x)\,$ (constant factor is omitted)
$a\,$	$0\,$	$ax\,$	$1\,$	$a^x\,$	$0\,$	$ax\,$	$1\,$	$a^x\,$
$x\,$	$1\,$	$\frac{x^2}{2}\,$	$\sqrt[x]{e}\,$	$\frac{x^x}{e^x}\,$	$1\,$	$\frac{x^2}{2}-\frac x2\,$	$1+\frac 1x\,$	$\Gamma(x)\,$
$ax+b\,$	$a\,$	$\frac{ax^2+2bx}{2}\,$	$\exp\left(\frac{a}{ax+b}\right)\,$	$\frac{(b+a x)^{\frac{b}{a}+x}}{e^x}\,$	$a\,$	$\frac{ax^2+2bx-ax}{2}\,$	$1+\frac{a}{ax+b}\,$	$\frac{a^x\Gamma(\frac{ax+b}{a})}{\Gamma(\frac{a+b}{a})}\,$
$\frac 1x\,$	$-\frac{1}{x^2}\,$	$\ln |x|\,$	$\frac{1}{\sqrt[x]{e}}\,$	$\frac{e^x}{x^x}\,$	$-\frac{1}{x+x^2}\,$	$\psi(x)\,$	$\frac{x}{x+1}\,$	$\frac{1}{\Gamma(x)}\,$
$x^a\,$	$ax^{a-1}\,$	$\frac{x^{a+1}}{a+1}\,$	$e^{\frac ax}\,$	$e^{-a x} x^{ax}\,$	$(x+1)^a-x^a\,$	$\frac{B_{a+1}(x)}{a+1},\,a\notin \mathbb{Z}^-\,$ $\frac{(-1)^{a-1}\psi^{(-a-1)}(x)}{\Gamma(-a)},\,a\in\mathbb{Z}^-$	$\left(1+\frac 1x\right)^a\,$	$\Gamma(x)^a\,$
$a^x\,$	$a^x\ln a\,$	$\frac{a^x}{\ln a}\,$	$a\,$	$a^{\frac{x^2}{2}}\,$	$(a-1)a^x\,$	$\frac{a^x}{a-1}\,$	$a\,$	$a^{\frac{x^2-x}{2}}\,$
$\sqrt[x]{a}\,$	$-\frac{\sqrt[x]{a}\ln a}{x^2}\,$	$x\sqrt[x]{a}-\operatorname{Ei}\left(\frac{\ln a}{x}\right)\ln a\,$	$a^{-\frac{1}{x^2}}\,$	$a^{\ln x}\,$	$a^{\frac{1}{1+x}}-a^{\frac{1}{x}}\,$	$? \,$	$a^{-\frac{1}{x+x^2}}\,$	$a^{\psi(x)}\,$
$\log_a x\,$	$\frac{1}{x \ln a}\,$	$\log_a x^x-\frac{x}{\ln a}$	$\exp \left(\frac{1}{x\ln x}\right) \,$	$\frac{(\log_a x)^x}{e^{\operatorname{li}(x)}}\,$	$\log_a\left(\frac 1x -1\right)\,$	$\log_a \Gamma(x)\,$	$\log_x (x+1)\,$	$?\,$
$x^x\,$	$x^x(1+\ln x)\,$	$?\,$	$ex\,$	$e^{-\frac{1}{4}x^2(1-2\ln x)}\,$	$(x+1)^{x+1}-x^x\,$	$? \,$	$\frac{(x+1)^{x+1}}{x^x}\,$	$\operatorname{K}(x)\,$
$\Gamma(x)\,$	$\Gamma(x)\psi(x)\,$	$?\,$	$e^{\psi(x)}\,$	$\frac{e^{\psi^{(-2)}(x)}}{\Gamma(x)}\,$	$(x-1)\Gamma(x)\,$	$(-1)^{x+1}\Gamma(x)(!(-x))\,$	$x\,$	$\frac{\Gamma(x)^{x-1}}{\operatorname{K}(x)}\,$

1. ^ H. R. Khatami & M. Jahanshahi & N. Aliev (2004). "An analytical method for some nonlinear difference equations by discrete multiplicative differentiation"., 5—10 July 2004, Antalya, Turkey – Dynamical Systems and Applications, Proceedings, pp. 455—462
2. ^ M. Jahanshahi, N. Aliev and H. R. Khatami (2004). "An analytic-numerical method for solving difference equations with variable coefficients by discrete multiplicative integration"., 5—10 July 2004, Antalya, Turkey – Dynamical Systems and Applications, Proceedings, pp. 425—435