Wichtige Differenziale

Lesedauer: 4 min | Vorlesen | Autor: Dr. Volkmar Naumburger

\( \frac{ {dy} }{ {dx} } = \frac{ {d({x^n})} }{ {dx} } = n \cdot {x^{n - 1} } \qquad n \in Q \) Gl. 40

Sonderfall:

\( y(x) = c; \quad \frac{ {dy} }{ {dx} } = \frac{ {d(c)} }{ {dx} } = \frac{ {d(c \cdot {x^0})} }{ {dx} } = 0 \cdot {x^{ - 1} } = 0 \)   mit c als Konstante Gl. 41

\(\frac{ {dy} }{ {dx} } = \frac{ {d({e^x})} }{ {dx} } = {e^x}\)   Exponentialfunktion, Basis e Gl. 42

\(\frac{ {dy} }{ {dx} } = \frac{ {d({a^x})} }{ {dx} } = {a^x} \cdot \ln a\)   Exponentialfunktion, beliebige Basis Gl. 43

\(\frac{ {dy} }{ {dx} } = \frac{ {d(\ln (x))} }{ {dx} } = \frac{1}{x}\)   natürlicher Logarithmus Gl. 44

\(\frac{ {dy} }{ {dx} } = \frac{ {d({ {\log }_a}(x))} }{ {dx} } = \frac{1}{x} \cdot {\log _a}e\)   Logarithmus beliebige Basis Gl. 45

\(\frac{ {dy} }{ {dx} } = \frac{ {d(\sin (x))} }{ {dx} } = \cos \left( x \right)\)   Winkelfunktionen Gl. 46

\(\frac{ {dy} }{ {dx} } = \frac{ {d(\cos (x))} }{ {dx} } = - \sin \left( x \right)\) Gl. 47

\(\frac{ {dy} }{ {dx} } = \frac{ {d(\tan (x))} }{ {dx} } = 1 + {\tan^2}\left( x \right)\) Gl. 48

\(\frac{ {dy} }{ {dx} } = \frac{ {d(\cot (x))} }{ {dx} } = - \left( {1 + { {\cot }^2}\left( x \right)} \right)\) Gl. 49

\(\frac{ {dy} }{ {dx} } = \frac{ {d(\arcsin (x))} }{ {dx} } = \frac{1}{ {\sqrt {1 - {x^2} } } }\)   zyklometrische Funktionen Gl. 50

\(\frac{ {dy} }{ {dx} } = \frac{ {d(\arccos (x))} }{ {dx} } = -\frac{1}{ {\sqrt {1 - {x^2} } } }\) Gl. 51

\(\frac{ {dy} }{ {dx} } = \frac{ {d(\arctan (x))} }{ {dx} } = \frac{1}{ {1 + {x^2} } }\) Gl. 52

\(\frac{ {dy} }{ {dx} } = \frac{ {d({\mathop{\rm arccot}\nolimits} (x))} }{ {dx} } = - \frac{1}{ {1 + {x^2} } }\) Gl. 53

\(\frac{ {dy} }{ {dx} } = \frac{ {d(\sinh (x))} }{ {dx} } = \cosh \left( x \right)\)   hyperbolische Funktionen Gl. 54

\(\frac{ {dy} }{ {dx} } = \frac{ {d(\cosh (x))} }{ {dx} } = \sinh \left( x \right)\) Gl. 55

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