Ableitungen von Grundfunktionen
| Funktion \(y\) | Ableitung \(\dot{y}\) | |
| Konstante Funktion | \(C =\) konstant | \(0\) |
| Potenzfunktionen | \(t^n\) | \(n \cdot t^{n-1}\) |
| \(\sqrt{t} = t^{\frac{1}{2}}\) | \(\dfrac{1}{2} \cdot t^{\frac{1}{2}-1} = \dfrac{1}{2} \cdot t^{-\frac{1}{2}} = \dfrac{1}{2\sqrt{t}}\) | |
| Trigonometrische Funktionen | \(\sin(t)\) | \(\cos(t)\) |
| \(\cos(t)\) | \(-\sin(t)\) | |
| \(\tan(t)\) | \(\dfrac{1}{\cos^2(t)}= 1 + \tan^2(t)\) | |
| \(\cot(t)\) | \(-\dfrac{1}{\sin^2(t)}= -1 - \cot^2(t)\) | |
| Arkusfunktionen | \(\sin^{-1}(t)\) | \(\dfrac{1}{\sqrt{1-t^2}}\) |
| \(\cos^{-1}(t)\) | \(-\dfrac{1}{\sqrt{1-t^2}}\) | |
| \(\tan^{-1}(t)\) | \(\dfrac{1}{1+t^2}\) | |
| \(\cot^{-1}(t)\) | \(-\dfrac{1}{1+t^2}\) | |
| Exponentialfunktionen | \(e^t\) | \(e^t\) |
| \(a^t\) | \(\ln(a) \cdot a^t\) | |
| Logarithmusfunktionen | \(\ln(t)\) | \(\dfrac{1}{t}\) |
| \(\log_a(t)\) | \(\dfrac{1}{\ln(a) \cdot t}\) | |
| Hyperbelfunktionen | \(\sinh(t)\) | \(\cosh(t)\) |
| \(\cosh(t)\) | \(\sinh(t)\) | |
| \(\tanh(t)\) | \(\dfrac{1}{\cosh^2(t)}= 1 - \tanh^2(t)\) | |
| \(\coth(t)\) | \(-\dfrac{1}{\sinh^2(t)}= 1 - \coth^2(t)\) | |
| Areafunktionen | \(\sinh^{-1}(t)\) | \(\dfrac{1}{\sqrt{t^2+1}}\) |
| \(\cosh^{-1}(t)\) | \(\dfrac{1}{\sqrt{t^2-1}}\) | |
| \(\tanh^{-1}(t)\) | \(\dfrac{1}{1-t^2}\) | |
| \(\coth^{-1}(t)\) | \(\dfrac{1}{1-t^2}\) | |