User:MathFacts/Alternative

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

In the following table $\psi (x,y)$ is the generalized polygamma function, $K(x)=e^{\zeta ^{\prime }(-1,x)-\zeta ^{\prime }(-1)}$ is the K-function, $(!x)={\frac {\Gamma (x+1,-1)}{e}}$ is subfactorial, $B_{n}(x)\,$ are the 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 derivative (multiplicative difference)	Discrete multiplicative integral (indefinite product) $\prod _{x}f(x)\,$ (constant factor is omitted)
$a\,$	$0\,$	$ax\,$	$1\,$	$a^{x}\,$	$0\,$	$ax\,$	$1\,$	$a^{x}\,$
$x\,$	$1\,$	${\frac {x^{2}}{2}}\,$	${\sqrt[{x}]{e}}\,$	${\frac {x^{x}}{e^{x}}}\,$	$1\,$	${\frac {x^{2}}{2}}-{\frac {x}{2}}\,$	$1+{\frac {1}{x}}\,$	$e^{\psi (-1,x)}\,$
$ax+b\,$	$a\,$	${\frac {ax^{2}+2bx}{2}}\,$	$\exp \left({\frac {a}{ax+b}}\right)\,$	${\frac {(b+ax)^{{\frac {b}{a}}+x}}{e^{x}}}\,$	$a\,$	${\frac {ax^{2}+2bx-ax}{2}}\,$	$1+{\frac {a}{ax+b}}\,$	$e^{x\ln \,a+\psi (-1,{\frac {ax+b}{a}})-\psi (-1,{\frac {a+b}{a}})}\,$
${\frac {1}{x}}\,$	$-{\frac {1}{x^{2}}}\,$	$\ln |x|\,$	${\frac {1}{\sqrt[{x}]{e}}}\,$	${\frac {e^{x}}{x^{x}}}\,$	$-{\frac {1}{x+x^{2}}}\,$	$\psi (0,x)\,$	${\frac {x}{x+1}}\,$	$e^{-\psi (-1,x)}\,$
$x^{a}\,$	$ax^{a-1}\,$	${\frac {x^{a+1}}{a+1}}\,$	$e^{\frac {a}{x}}\,$	$e^{-ax}x^{ax}\,$	$(x+1)^{a}-x^{a}\,$	$-{\frac {e^{\psi (-1,a+1)}\left(2^{a}\left(\psi \left(-a-1,{\frac {x}{2}}+{\frac {1}{2}}\right)+\psi \left(-a-1,{\frac {x}{2}}\right)\right)-\psi (-a-1,x)\right)}{\ln(2)}}$	$\left(1+{\frac {1}{x}}\right)^{a}\,$	$e^{a\psi (-1,x)}\,$
$a^{x}\,$	$a^{x}\ln a\,$	${\frac {a^{x}}{\ln a}}\,$	$a\,$	$a^{\frac {x^{2}}{2}}\,$	$(a-1)a^{x}\,$	${\frac {a^{x}}{a-1}}\,$	$a\,$	$a^{\frac {x^{2}-x}{2}}\,$
${\sqrt[{x}]{a}}\,$	$-{\frac {{\sqrt[{x}]{a}}\ln a}{x^{2}}}\,$	$x{\sqrt[{x}]{a}}-\operatorname {Ei} \left({\frac {\ln a}{x}}\right)\ln a\,$	$a^{-{\frac {1}{x^{2}}}}\,$	$a^{\ln x}\,$	$a^{\frac {1}{1+x}}-a^{\frac {1}{x}}\,$	$?\,$	$a^{-{\frac {1}{x+x^{2}}}}\,$	$a^{\psi (0,x)}\,$
$\log _{a}x\,$	${\frac {1}{x\ln a}}\,$	$\log _{a}x^{x}-{\frac {x}{\ln a}}$	$\exp \left({\frac {1}{x\ln x}}\right)\,$	${\frac {(\log _{a}x)^{x}}{e^{\operatorname {li} (x)}}}\,$	$\log _{a}\left({\frac {1}{x}}-1\right)\,$	${\frac {\psi (-1,x)}{\ln a}}\,$	$\log _{x}(x+1)\,$	$?\,$
$x^{x}\,$	$x^{x}(1+\ln x)\,$	$?\,$	$ex\,$	$e^{x^{2}-2x^{2}\ln x}\,$	$(x+1)^{x+1}-x^{x}\,$	$?\,$	${\frac {(x+1)^{x+1}}{x^{x}}}\,$	$e^{\psi (-2,x)+{\frac {x^{2}-x}{2}}}\,$
$\Gamma (x)\,$	$\Gamma (x)\psi (0,x)\,$	$?\,$	$e^{\psi (0,x)}\,$	$e^{\psi (-2,x)-{\frac {x}{2}}\ln \,2\pi }\,$	$(x-1)\Gamma (x)\,$	$(-1)^{x+1}\Gamma (x)(!(-x))\,$	$x\,$	$e^{(x-1)\psi (-1,x)-\psi (-2,x)-{\frac {x^{2}-x}{2}}}\,$