d/(dt)((4e^t)/((4-3e^t)^2))

Lösung

\frac{d}{d t} (\frac{4 e ^{t}}{( 4 - 3 e ^{t} ) ^{2}})

\frac{4 e ^{t} ( 3 e ^{t} + 4 )}{( 4 - 3 e ^{t} ) ^{3}}

Schritte zur Lösung

\frac{d}{d t} (\frac{4 e ^{t}}{( 4 - 3 e ^{t} ) ^{2}})

Entferne die Konstante:

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

= 4 \frac{d}{d t} (\frac{e ^{t}}{( 4 - 3 e ^{t} ) ^{2}})

Wende die Quotientenregel an:

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

= 4 \cdot \frac{\frac{d}{d t} ( e ^{t} ) ( 4 - 3 e ^{t} ) ^{2} - \frac{d}{d t} ( ( 4 - 3 e ^{t} ) ^{2} ) e ^{t}}{( ( 4 - 3 e ^{t} ) ^{2} ) ^{2}}

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

\frac{d}{d t} ((4 - 3 e^{t})^{2}) = - 6 e^{t} (4 - 3 e^{t})

= 4 \cdot \frac{e ^{t} ( 4 - 3 e ^{t} ) ^{2} - ( - 6 e ^{t} ( 4 - 3 e ^{t} ) ) e ^{t}}{( ( 4 - 3 e ^{t} ) ^{2} ) ^{2}}

Vereinfache

4 \cdot \frac{e ^{t} ( 4 - 3 e ^{t} ) ^{2} - ( - 6 e ^{t} ( 4 - 3 e ^{t} ) ) e ^{t}}{( ( 4 - 3 e ^{t} ) ^{2} ) ^{2}} : \frac{4 e ^{t} ( 3 e ^{t} + 4 )}{( 4 - 3 e ^{t} ) ^{3}}

= \frac{4 e ^{t} ( 3 e ^{t} + 4 )}{( 4 - 3 e ^{t} ) ^{3}}

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