derivative-(40(9+5x^2))/((9-5x^2)^2)

Lösung

ableitung von

- \frac{40 ( 9 + 5 x ^{2} )}{( 9 - 5 x ^{2} ) ^{2}}

- \frac{400 x ( 5 x ^{2} + 27 )}{( 9 - 5 x ^{2} ) ^{3}}

Schritte zur Lösung

\frac{d}{d x} (- \frac{40 ( 9 + 5 x ^{2} )}{( 9 - 5 x ^{2} ) ^{2}})

Entferne die Konstante:

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

= - 40 \frac{d}{d x} (\frac{9 + 5 x ^{2}}{( 9 - 5 x ^{2} ) ^{2}})

Wende die Quotientenregel an:

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

= - 40 \cdot \frac{\frac{d}{d x} ( 9 + 5 x ^{2} ) ( 9 - 5 x ^{2} ) ^{2} - \frac{d}{d x} ( ( 9 - 5 x ^{2} ) ^{2} ) ( 9 + 5 x ^{2} )}{( ( 9 - 5 x ^{2} ) ^{2} ) ^{2}}

\frac{d}{d x} (9 + 5 x^{2}) = 10 x

\frac{d}{d x} ((9 - 5 x^{2})^{2}) = - 20 x (9 - 5 x^{2})

= - 40 \cdot \frac{10 x ( 9 - 5 x ^{2} ) ^{2} - ( - 20 x ( 9 - 5 x ^{2} ) ) ( 9 + 5 x ^{2} )}{( ( 9 - 5 x ^{2} ) ^{2} ) ^{2}}

Vereinfache

- 40 \cdot \frac{10 x ( 9 - 5 x ^{2} ) ^{2} - ( - 20 x ( 9 - 5 x ^{2} ) ) ( 9 + 5 x ^{2} )}{( ( 9 - 5 x ^{2} ) ^{2} ) ^{2}} : - \frac{400 x ( 5 x ^{2} + 27 )}{( 9 - 5 x ^{2} ) ^{3}}

= - \frac{400 x ( 5 x ^{2} + 27 )}{( 9 - 5 x ^{2} ) ^{3}}

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