d/(dt)((ce^t)/(1+ce^t))

Lösung

\frac{d}{d t} (\frac{c e ^{t}}{1 + c e ^{t}})

\frac{c e ^{t}}{( 1 + c e ^{t} ) ^{2}}

Schritte zur Lösung

\frac{d}{d t} (\frac{c e ^{t}}{1 + c e ^{t}})

Entferne die Konstante:

(a \cdot f)^{'} = a \cdot f^{'}

= c \frac{d}{d t} (\frac{e ^{t}}{1 + c e ^{t}})

Wende die Quotientenregel an:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= c \frac{\frac{d}{d t} ( e ^{t} ) ( 1 + c e ^{t} ) - \frac{d}{d t} ( 1 + c e ^{t} ) e ^{t}}{( 1 + c e ^{t} ) ^{2}}

\frac{d}{d t} (e^{t}) = e^{t}

\frac{d}{d t} (1 + c e^{t}) = c e^{t}

= c \frac{e ^{t} ( 1 + c e ^{t} ) - c e ^{t} e ^{t}}{( 1 + c e ^{t} ) ^{2}}

Vereinfache

c \frac{e ^{t} ( 1 + c e ^{t} ) - c e ^{t} e ^{t}}{( 1 + c e ^{t} ) ^{2}} : \frac{c e ^{t}}{( 1 + c e ^{t} ) ^{2}}

= \frac{c e ^{t}}{( 1 + c e ^{t} ) ^{2}}