derivative (x^3)/(5-x^2)

Lösung

ableitung von

\frac{x ^{3}}{5 - x ^{2}}

\frac{15 x ^{2} - x ^{4}}{( 5 - x ^{2} ) ^{2}}

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

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

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

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

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

Vereinfache

\frac{3 x ^{2} ( 5 - x ^{2} ) - ( - 2 x ) x ^{3}}{( 5 - x ^{2} ) ^{2}} : \frac{15 x ^{2} - x ^{4}}{( 5 - x ^{2} ) ^{2}}

= \frac{15 x ^{2} - x ^{4}}{( 5 - x ^{2} ) ^{2}}

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