Es gibt noch viel zu lernen ... und es wird mit jedem Tag mehr.
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} mit 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}
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