derivative f(x)=(x(x^2+4))/(x-2)

Lösung

ableitung von

f (x) = \frac{x ( x ^{2} + 4 )}{x - 2}

\frac{2 x ^{3} - 6 x ^{2} - 8}{( x - 2 ) ^{2}}

Schritte zur Lösung

\frac{d}{d x} (\frac{x ( x ^{2} + 4 )}{x - 2})

Wende die Quotientenregel an:

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

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

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

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

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

Vereinfache

\frac{( 3 x ^{2} + 4 ) ( x - 2 ) - 1 \cdot x ( x ^{2} + 4 )}{( x - 2 ) ^{2}} : \frac{2 x ^{3} - 6 x ^{2} - 8}{( x - 2 ) ^{2}}

= \frac{2 x ^{3} - 6 x ^{2} - 8}{( x - 2 ) ^{2}}

\int 7 x^{- \frac{8}{9}} d x

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