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1.
d
d
x
(
k
)
=
0
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}(k)=0}
2.
d
d
x
(
x
)
=
1
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}(x)=1}
3.
d
d
x
(
x
n
)
=
n
⋅
x
n
−
1
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}(x^{n})=n\cdot x^{n-1}}
4.
d
d
x
(
sin
(
x
)
)
=
cos
(
x
)
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}(\sin(x))=\cos(x)}
5.
d
d
x
(
cos
(
x
)
)
=
−
sin
(
x
)
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}(\cos(x))=-\sin(x)}
6.
d
d
x
(
tan
(
x
)
)
=
1
cos
2
(
x
)
=
1
+
tan
2
(
x
)
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}(\tan(x))={\frac {1}{\cos ^{2}(x)}}=1+\tan ^{2}(x)}
7.
d
d
x
(
cot
(
x
)
)
=
1
−
sin
2
(
x
)
=
−
(
1
+
cot
2
(
x
)
)
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}(\cot(x))={\frac {1}{-\sin ^{2}(x)}}=-(1+\cot ^{2}(x))}
8.
d
d
x
(
e
x
)
=
e
x
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}(e^{x})=e^{x}\,}
9.
d
d
x
(
a
x
)
=
a
x
⋅
ln
(
a
)
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}(a^{x})=a^{x}\cdot \ln(a)}
10.
d
d
x
(
ln
(
x
)
)
=
1
x
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}(\ln(x))={\frac {1}{x}}}
11.
d
d
x
(
a
log
(
x
)
)
=
1
x
⋅
1
ln
(
a
)
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}(\,^{a}\log(x))={\frac {1}{x}}\cdot {\frac {1}{\ln(a)}}}
12.
d
d
x
(
arcsin
(
x
)
)
=
1
1
−
x
2
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}(\arcsin(x))={\frac {1}{\sqrt {1-x^{2}}}}}
13.
d
d
x
(
arccos
(
x
)
)
=
−
1
1
−
x
2
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}(\arccos(x))={\frac {-1}{\sqrt {1-x^{2}}}}}
14.
d
d
x
(
arctan
(
x
)
)
=
1
1
+
x
2
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}(\arctan(x))={\frac {1}{1+x^{2}}}}
15.
d
d
x
(
sinh
(
x
)
)
=
cosh
(
x
)
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}(\sinh(x))=\cosh(x)}
16.
d
d
x
(
cosh
(
x
)
)
=
sinh
(
x
)
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}(\cosh(x))=\sinh(x)}
17.
d
d
x
(
tanh
(
x
)
)
=
1
cosh
2
(
x
)
=
1
−
tanh
(
x
)
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}(\tanh(x))={\frac {1}{\cosh ^{2}(x)}}=1-\tanh(x)}
18.
d
d
x
(
coth
(
x
)
)
=
−
1
sinh
2
(
x
)
=
1
−
coth
(
x
)
{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}(\coth(x))={\frac {-1}{\sinh ^{2}(x)}}=1-\coth(x)}
1.
∫
0
d
x
=
C
{\displaystyle \int 0\;\mathrm {d} x=C}
2.
∫
1
d
x
=
x
+
C
{\displaystyle \int 1\;\mathrm {d} x=x+C}
3.
∫
k
d
x
=
k
x
+
C
{\displaystyle \int k\;\mathrm {d} x=kx+C}
4.
∫
x
n
d
x
=
x
n
+
1
n
+
1
+
C
{\displaystyle \int x^{n}\;\mathrm {d} x={\frac {x^{n+1}}{n+1}}+C}
n
∈
R
∖
{
−
1
}
{\displaystyle n\in \mathbb {R} \setminus \{-1\}}
x
>
0
{\displaystyle x>0\,}
5.
∫
x
−
1
d
x
=
ln
|
x
|
+
C
{\displaystyle \int x^{-1}\;\mathrm {d} x=\ln |x|+C}
6.
∫
sin
(
x
)
d
x
=
−
cos
(
x
)
+
C
{\displaystyle \int \sin(x)\;\mathrm {d} x=-\cos(x)+C}
7.
∫
cos
(
x
)
d
x
=
sin
(
x
)
+
C
{\displaystyle \int \cos(x)\;\mathrm {d} x=\sin(x)+C}
8.
∫
tan
(
x
)
d
x
=
−
ln
|
cos
(
x
)
|
+
C
{\displaystyle \int \tan(x)\;\mathrm {d} x=-\ln |\cos(x)|+C}
9.
∫
cot
(
x
)
d
x
=
ln
|
sin
(
x
)
|
+
C
{\displaystyle \int \cot(x)\;\mathrm {d} x=\ln |\sin(x)|+C}
10.
∫
d
x
cos
2
(
x
)
=
tan
(
x
)
+
C
{\displaystyle \int {\frac {\mathrm {d} x}{\cos ^{2}(x)}}=\tan(x)+C}
11.
∫
d
x
sin
2
(
x
)
=
−
cot
(
x
)
+
C
{\displaystyle \int {\frac {\mathrm {d} x}{\sin ^{2}(x)}}=-\cot(x)+C}
12.
∫
e
x
d
x
=
e
x
+
C
{\displaystyle \int e^{x}\;\mathrm {d} x=e^{x}+C}
13.
∫
a
x
d
x
=
a
x
ln
(
a
)
+
C
{\displaystyle \int a^{x}\;\mathrm {d} x={\frac {a^{x}}{\ln(a)}}+C}
14.
∫
ln
(
x
)
d
x
=
x
⋅
ln
(
x
)
−
x
+
C
{\displaystyle \int \ln(x)\;\mathrm {d} x=x\cdot \ln(x)-x+C}
15.
∫
a
log
(
x
)
d
x
=
1
ln
(
a
)
⋅
(
x
⋅
ln
(
x
)
−
x
)
+
C
{\displaystyle \int \,^{a}\log(x)\;\mathrm {d} x={\frac {1}{\ln(a)}}\cdot (x\cdot \ln(x)-x)+C}
16.
∫
1
1
−
x
2
d
x
=
arcsin
(
x
)
+
C
{\displaystyle \int {\frac {1}{\sqrt {1-x^{2}}}}\;\mathrm {d} x=\arcsin(x)+C}
17.
∫
−
1
1
−
x
2
d
x
=
arccos
(
x
)
+
C
{\displaystyle \int {\frac {-1}{\sqrt {1-x^{2}}}}\;\mathrm {d} x=\arccos(x)+C}
18.
∫
1
1
+
x
2
d
x
=
arctan
(
x
)
+
C
{\displaystyle \int {\frac {1}{1+x^{2}}}\;\mathrm {d} x=\arctan(x)+C}
19.
∫
sinh
(
x
)
d
x
=
cosh
(
x
)
+
C
{\displaystyle \int \sinh(x)\;\mathrm {d} x=\cosh(x)+C}
20.
∫
cosh
(
x
)
d
x
=
sinh
(
x
)
+
C
{\displaystyle \int \cosh(x)\;\mathrm {d} x=\sinh(x)+C}
21.
∫
tanh
(
x
)
d
x
=
ln
(
cosh
(
x
)
)
+
C
{\displaystyle \int \tanh(x)\;\mathrm {d} x=\ln(\cosh(x))+C}
22.
∫
coth
(
x
)
d
x
=
ln
|
sinh
(
x
)
|
+
C
{\displaystyle \int \coth(x)\;\mathrm {d} x=\ln |\sinh(x)|+C}
