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