Differentiationsregeln

Einige Differentiationsregeln sind



Definition der Ableitung $ u = f(t)$
$ u' = \frac{d u}{dt} = f'(t) =$
$ \lim\limits_{\Delta t\rightarrow 0} \frac{f(t+\Delta t)-f(t)}{\Delta t}
$
     
Partielle Ableitung $ u = f(x,y,z,\ldots,t)$
$ \frac{\partial u}{\partial x} =$
$ \lim\limits_{\Delta x
\rightarrow
0} \frac{f(x+\Delta x, y, z,\ldots,t)-f(x,y,z,\ldots,t)}{\Delta x}$
     
Andere Schreibweise $ u = f(t)$ $ \frac{d u}{dt} = \frac{d}{dt} u = \frac{d}{dt} f(t) $
     
Konstanter Faktor $ u = f(x),\; c=const$ $ \frac{d cu}{dx} = c \frac{du}{dx}$
     
Summenregel $ u = f(t),\; v = g(t)$ $ \frac{d(u+v)}{dt} = \frac{d u}{dt} + \frac{d v}{dt}$
     
Produktregel $ u = f(t),\; v = g(t)$ $ \frac{d u v}{dt} = u \frac{dv}{dt}+v\frac{du}{dt}$
     
Bruch $ u = f(t),\; v = g(t)$ $ \frac{d}{dt}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dt}-u
\frac{dv}{dt}}{v^2}$
     
Kettenregel $ u = f(v),\; v = g(t)$ $ \frac{d f(g(t))}{dt} = \frac{d f(v}{d v}\frac{d v}{dt} = \frac{d f(v}{d v}\frac{d g(t)}{dt}
$
     
logarithmische Ableitung $ u = f(x)$ $ \frac{d \ln u}{dx} = \frac{\frac{d y}{dx}}{y}$
Differentiationsregeln















Othmar Marti
Experimentelle Physik
Universiät Ulm