# Wichtige Differenziale

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