Funkcja
Pochodna
Uwagi
c
{\displaystyle c\,}
0
{\displaystyle 0\,}
x
{\displaystyle x\,}
1
{\displaystyle 1\,}
x
n
{\displaystyle x^{n}\,}
n
x
n
−
1
{\displaystyle nx^{n-1}\,}
n
∈
R
∖
{
1
}
4
)
{\displaystyle n\in \mathbb {R} \setminus \{1\}\qquad ^{4)}}
a
x
+
b
{\displaystyle ax+b\,}
a
{\displaystyle a\,}
a
x
2
+
b
x
+
c
{\displaystyle ax^{2}+bx+c\,}
2
a
x
+
b
{\displaystyle 2ax+b\,}
a
x
=
a
⋅
x
−
1
{\displaystyle {a \over x}=a\cdot x^{-1}}
−
a
x
2
=
−
1
⋅
a
⋅
x
−
2
{\displaystyle -{a \over x^{2}}=-1\cdot a\cdot x^{-2}\,}
x
≠
0
{\displaystyle x\neq 0\,}
sin
x
{\displaystyle \sin x\,}
cos
x
{\displaystyle \cos x\,}
cos
x
{\displaystyle \cos x\,}
−
sin
x
{\displaystyle -\sin x\,}
tg
x
{\displaystyle \operatorname {tg} \ x\,}
sec
2
x
=
1
cos
2
x
{\displaystyle \operatorname {sec} ^{2}\ x={1 \over \cos ^{2}x}\,}
x
≠
π
2
+
k
π
,
k
∈
Z
{\displaystyle x\neq {\pi \over 2}+k\pi ,\;k\in \mathbb {Z} }
ctg
x
{\displaystyle \operatorname {ctg} \ x\,}
−
csc
2
x
=
−
1
sin
2
x
{\displaystyle -\operatorname {csc} ^{2}\ x=-{1 \over \sin ^{2}x}\,}
x
≠
k
π
,
k
∈
Z
{\displaystyle x\not =k\pi ,\;k\in \mathbb {Z} }
sec
x
{\displaystyle \operatorname {sec} \ x\,}
tg
x
sec
x
{\displaystyle \operatorname {tg} \ x\ \operatorname {sec} \ x\,}
x
≠
π
2
+
k
π
,
k
∈
Z
{\displaystyle x\neq {\pi \over 2}+k\pi ,\;k\in \mathbb {Z} }
csc
x
{\displaystyle \operatorname {csc} \ x\,}
−
ctg
x
csc
x
{\displaystyle -\operatorname {ctg} \ x\ \operatorname {csc} \ x\,}
x
≠
k
π
,
k
∈
Z
{\displaystyle x\not =k\pi ,\;k\in \mathbb {Z} }
e
x
{\displaystyle e^{x}\,}
e
x
{\displaystyle e^{x}\,}
a
x
{\displaystyle a^{x}\,}
a
x
ln
a
{\displaystyle a^{x}\ln a\,}
a
>
0
{\displaystyle a>0\,}
x
x
{\displaystyle x^{x}\,}
x
x
(
1
+
ln
x
)
{\displaystyle x^{x}(1+\ln x)\,}
x
>
0
{\displaystyle x>0\,}
ln
x
{\displaystyle \ln x\,}
1
x
{\displaystyle 1 \over x\,}
x
>
0
{\displaystyle x>0\,}
log
a
x
{\displaystyle \log _{a}x\,}
1
x
ln
a
{\displaystyle 1 \over x\ln a\,}
a
r
c
s
i
n
x
{\displaystyle \operatorname {arc\,sin} \ x\,}
1
1
−
x
2
{\displaystyle 1 \over {\sqrt {1-x^{2}}}\,}
|
x
|
<
1
{\displaystyle |x|<1\,}
a
r
c
c
o
s
x
{\displaystyle \operatorname {arc\,cos} \ x\,}
−
1
1
−
x
2
{\displaystyle -{1 \over {\sqrt {1-x^{2}}}}\,}
|
x
|
<
1
{\displaystyle |x|<1\,}
a
r
c
t
g
x
{\displaystyle \operatorname {arc\,tg} \ x\,}
1
1
+
x
2
{\displaystyle 1 \over 1+x^{2}\,}
a
r
c
c
t
g
x
{\displaystyle \operatorname {arc\,ctg} \ x\,}
−
1
1
+
x
2
{\displaystyle -{1 \over 1+x^{2}}\,}
a
r
c
s
e
c
x
{\displaystyle \operatorname {arc\,sec} \ x\,}
1
x
2
1
−
1
x
2
{\displaystyle {\frac {1}{x^{2}{\sqrt {1-{\frac {1}{x^{2}}}}}}}\,}
|
x
|
>
1
{\displaystyle |x|>1\,}
a
r
c
c
s
c
x
{\displaystyle \operatorname {arc\,csc} \ x\,}
−
1
x
2
1
−
1
x
2
{\displaystyle -{\frac {1}{x^{2}{\sqrt {1-{\frac {1}{x^{2}}}}}}}\,}
|
x
|
>
1
{\displaystyle |x|>1\,}
x
{\displaystyle {\sqrt {x}}\,}
1
2
x
{\displaystyle 1 \over 2{\sqrt {x}}\,}
x
>
0
{\displaystyle x>0\,}
x
n
{\displaystyle {\sqrt[{n}]{x}}\,}
1
n
x
n
−
1
n
{\displaystyle 1 \over n{\sqrt[{n}]{x^{n-1}}}\,}
x
>
0
{\displaystyle x>0\,}
sinh
x
=
e
x
−
e
−
x
2
{\displaystyle \operatorname {sinh} \ x={{e^{x}-e^{-x}} \over 2}\,}
cosh
x
=
e
x
+
e
−
x
2
{\displaystyle \operatorname {cosh} \ x={{e^{x}+e^{-x}} \over 2}\,}
cosh
x
=
e
x
+
e
−
x
2
{\displaystyle \operatorname {cosh} \ x={{e^{x}+e^{-x}} \over 2}\,}
sinh
x
=
e
x
−
e
−
x
2
{\displaystyle \operatorname {sinh} \ x={{e^{x}-e^{-x}} \over 2}\,}
tgh
x
=
sinh
x
cosh
x
{\displaystyle \operatorname {tgh} \ x={\operatorname {sinh} \ x \over \operatorname {cosh} \ x}\,}
sech
2
x
=
1
cosh
2
x
=
4
(
e
x
+
e
−
x
)
2
{\displaystyle \operatorname {sech} ^{2}\ x={\tfrac {1}{\operatorname {cosh} ^{2}\ x}}={\tfrac {4}{(e^{x}+e^{-x})^{2}}}\,}
ctgh
x
=
cosh
x
sinh
x
{\displaystyle \operatorname {ctgh} \ x={\frac {\operatorname {cosh} \ x}{\operatorname {sinh} \ x}}\,}
−
csch
2
x
=
−
1
sinh
2
x
=
−
4
(
e
x
−
e
−
x
)
2
{\displaystyle -\operatorname {csch} ^{2}\ x=-{\tfrac {1}{\operatorname {sinh} ^{2}\ x}}=-{\tfrac {4}{(e^{x}-e^{-x})^{2}}}\,}
x
≠
0
{\displaystyle x\neq 0\,}
sech
x
=
2
e
x
+
e
−
x
{\displaystyle \operatorname {sech} \ x={\frac {2}{e^{x}+e^{-x}}}\,}
−
tgh
x
sech
x
{\displaystyle -\operatorname {tgh} \ x\ \operatorname {sech} \ x\,}
csch
x
=
2
e
x
−
e
−
x
{\displaystyle \operatorname {csch} \ x={\frac {2}{e^{x}-e^{-x}}}\,}
−
ctgh
x
csch
x
{\displaystyle -\operatorname {ctgh} \ x\ \operatorname {csch} \ x\,}
x
≠
0
{\displaystyle x\neq 0\,}
a
r
s
i
n
h
x
=
ln
(
x
+
x
2
+
1
)
{\displaystyle \operatorname {ar\,sinh} \ x=\ln(x+{\sqrt {x^{2}+1}})\,}
1
x
2
+
1
{\displaystyle {\frac {1}{\sqrt {x^{2}+1}}}\,}
a
r
c
o
s
h
x
=
ln
(
x
+
x
−
1
x
+
1
)
{\displaystyle \operatorname {ar\,cosh} \ x=\ln(x+{\sqrt {x-1}}{\sqrt {x+1}})\,}
1
x
2
−
1
{\displaystyle {\frac {1}{\sqrt {x^{2}-1}}}\,}
x
>
1
{\displaystyle x>1\,}
a
r
t
g
h
x
=
1
2
ln
1
+
x
1
−
x
{\displaystyle \operatorname {ar\,tgh} \ x={\frac {1}{2}}\ln {1+x \over 1-x}\,}
1
1
−
x
2
{\displaystyle 1 \over 1-x^{2}\,}
|
x
|
<
1
{\displaystyle |x|<1\,}
a
r
c
t
g
h
x
=
1
2
ln
x
+
1
x
−
1
{\displaystyle \operatorname {ar\,ctgh} \ x={\frac {1}{2}}\ln {x+1 \over x-1}\,}
1
1
−
x
2
{\displaystyle 1 \over 1-x^{2}\,}
|
x
|
>
1
{\displaystyle |x|>1\,}
a
r
s
e
c
h
x
=
ln
(
1
x
−
1
1
x
+
1
+
1
x
)
{\displaystyle \operatorname {ar\,sech} \ x=\ln \left({\sqrt {{\tfrac {1}{x}}-1}}{\sqrt {{\tfrac {1}{x}}+1}}+{\tfrac {1}{x}}\right)\,}
−
1
x
(
x
+
1
)
1
−
x
1
+
x
5
)
{\displaystyle {\frac {-1}{x(x+1)\,{\sqrt {\frac {1-x}{1+x}}}}}\qquad ^{5)}\,}
x
∈
(
0
;
1
)
{\displaystyle x\in (0;1)\,}
a
r
c
s
c
h
x
=
ln
(
1
+
1
x
2
+
1
x
)
{\displaystyle \operatorname {ar\,csch} \ x=\ln \left({\sqrt {1+{\tfrac {1}{x^{2}}}}}+{\tfrac {1}{x}}\right)\,}
−
1
x
2
1
+
1
x
2
5
)
{\displaystyle {\frac {-1}{x^{2}\,{\sqrt {1+{\frac {1}{x^{2}}}}}}}\qquad ^{5)}\,}
x
≠
0
{\displaystyle x\neq 0\,}
ln
(
x
+
x
2
±
a
2
)
{\displaystyle \ln(x+{\sqrt {x^{2}\pm a^{2}}})\,}
1
x
2
±
a
2
{\displaystyle 1 \over {\sqrt {x^{2}\pm a^{2}}}\,}
![{\displaystyle {\sqrt[{n}]{x}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d7be8627e12afe7ab3dc3f7f4229f24141864e8)
![{\displaystyle 1 \over n{\sqrt[{n}]{x^{n-1}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53689e1ec0d8f25356f4574e4f854ad91145d6dd)
