derivative (3x^{3/5}-x)/(x-1)

Lösung

ableitung von

\frac{3 x ^{\frac{3}{5}} - x}{x - 1}

\frac{- 6 x - 9 + 5 x ^{\frac{2}{5}}}{5 x ^{\frac{2}{5}} ( x - 1 ) ^{2}}

Schritte zur Lösung

\frac{d}{d x} (\frac{3 x ^{\frac{3}{5}} - x}{x - 1})

Wende die Quotientenregel an:

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

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

\frac{d}{d x} (3 x^{\frac{3}{5}} - x) = \frac{9}{5 x ^{\frac{2}{5}}} - 1

\frac{d}{d x} (x - 1) = 1

= \frac{( \frac{9}{5 x ^{\frac{2}{5}}} - 1 ) ( x - 1 ) - 1 \cdot ( 3 x ^{\frac{3}{5}} - x )}{( x - 1 ) ^{2}}

Vereinfache

\frac{( \frac{9}{5 x ^{\frac{2}{5}}} - 1 ) ( x - 1 ) - 1 \cdot ( 3 x ^{\frac{3}{5}} - x )}{( x - 1 ) ^{2}} : \frac{- 6 x - 9 + 5 x ^{\frac{2}{5}}}{5 x ^{\frac{2}{5}} ( x - 1 ) ^{2}}

= \frac{- 6 x - 9 + 5 x ^{\frac{2}{5}}}{5 x ^{\frac{2}{5}} ( x - 1 ) ^{2}}

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