derivative (2x^2)/((3-2x)^4)

Lösung

ableitung von

\frac{2 x ^{2}}{( 3 - 2 x ) ^{4}}

\frac{4 x ( 2 x + 3 )}{( 3 - 2 x ) ^{5}}

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Wende die Quotientenregel an:

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

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

\frac{d}{d x} (x^{2}) = 2 x

\frac{d}{d x} ((3 - 2 x)^{4}) = - 8 (3 - 2 x)^{3}

= 2 \cdot \frac{2 x ( 3 - 2 x ) ^{4} - ( - 8 ( 3 - 2 x ) ^{3} ) x ^{2}}{( ( 3 - 2 x ) ^{4} ) ^{2}}

Vereinfache

2 \cdot \frac{2 x ( 3 - 2 x ) ^{4} - ( - 8 ( 3 - 2 x ) ^{3} ) x ^{2}}{( ( 3 - 2 x ) ^{4} ) ^{2}} : \frac{4 x ( 2 x + 3 )}{( 3 - 2 x ) ^{5}}

= \frac{4 x ( 2 x + 3 )}{( 3 - 2 x ) ^{5}}

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