derivative of (2x^5+3x^4-1/(-2x^7))

Lösung

\frac{d}{d x} (\frac{2 x ^{5} + 3 x ^{4} - 1}{- 2 x ^{7}})

- \frac{- 4 x ^{5} - 9 x ^{4} + 7}{2 x ^{8}}

Schritte zur Lösung

\frac{d}{d x} (\frac{2 x ^{5} + 3 x ^{4} - 1}{- 2 x ^{7}})

Vereinfache

\frac{2 x ^{5} + 3 x ^{4} - 1}{- 2 x ^{7}} : - \frac{2 x ^{5} + 3 x ^{4} - 1}{2 x ^{7}}

= \frac{d}{d x} (- \frac{2 x ^{5} + 3 x ^{4} - 1}{2 x ^{7}})

Entferne die Konstante:

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

= - \frac{1}{2} \frac{d}{d x} (\frac{2 x ^{5} + 3 x ^{4} - 1}{x ^{7}})

Wende die Quotientenregel an:

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

= - \frac{1}{2} \cdot \frac{\frac{d}{d x} ( 2 x ^{5} + 3 x ^{4} - 1 ) x ^{7} - \frac{d}{d x} ( x ^{7} ) ( 2 x ^{5} + 3 x ^{4} - 1 )}{( x ^{7} ) ^{2}}

\frac{d}{d x} (2 x^{5} + 3 x^{4} - 1) = 10 x^{4} + 12 x^{3}

\frac{d}{d x} (x^{7}) = 7 x^{6}

= - \frac{1}{2} \cdot \frac{( 10 x ^{4} + 12 x ^{3} ) x ^{7} - 7 x ^{6} ( 2 x ^{5} + 3 x ^{4} - 1 )}{( x ^{7} ) ^{2}}

Vereinfache

- \frac{1}{2} \cdot \frac{( 10 x ^{4} + 12 x ^{3} ) x ^{7} - 7 x ^{6} ( 2 x ^{5} + 3 x ^{4} - 1 )}{( x ^{7} ) ^{2}} : - \frac{- 4 x ^{5} - 9 x ^{4} + 7}{2 x ^{8}}

= - \frac{- 4 x ^{5} - 9 x ^{4} + 7}{2 x ^{8}}

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