derivative of (x^4/(5-x^3))

Lösung

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

\frac{20 x ^{3} - x ^{6}}{( 5 - x ^{3} ) ^{2}}

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

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

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

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

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

Vereinfache

\frac{4 x ^{3} ( 5 - x ^{3} ) - ( - 3 x ^{2} ) x ^{4}}{( 5 - x ^{3} ) ^{2}} : \frac{20 x ^{3} - x ^{6}}{( 5 - x ^{3} ) ^{2}}

= \frac{20 x ^{3} - x ^{6}}{( 5 - x ^{3} ) ^{2}}