derivative (x^6)/(5-x^5)

Lösung

ableitung von

\frac{x ^{6}}{5 - x ^{5}}

\frac{30 x ^{5} - x ^{10}}{( 5 - x ^{5} ) ^{2}}

Schritte zur Lösung

\frac{d}{d x} (\frac{x ^{6}}{5 - x ^{5}})

Wende die Quotientenregel an:

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

= \frac{\frac{d}{d x} ( x ^{6} ) ( 5 - x ^{5} ) - \frac{d}{d x} ( 5 - x ^{5} ) x ^{6}}{( 5 - x ^{5} ) ^{2}}

\frac{d}{d x} (x^{6}) = 6 x^{5}

\frac{d}{d x} (5 - x^{5}) = - 5 x^{4}

= \frac{6 x ^{5} ( 5 - x ^{5} ) - ( - 5 x ^{4} ) x ^{6}}{( 5 - x ^{5} ) ^{2}}

Vereinfache

\frac{6 x ^{5} ( 5 - x ^{5} ) - ( - 5 x ^{4} ) x ^{6}}{( 5 - x ^{5} ) ^{2}} : \frac{30 x ^{5} - x ^{10}}{( 5 - x ^{5} ) ^{2}}

= \frac{30 x ^{5} - x ^{10}}{( 5 - x ^{5} ) ^{2}}

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