derivative f(x)=(2-5x-x^2)/(x^2-4)

Lösung

ableitung von

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

\frac{5 x ^{2} + 4 x + 20}{( x ^{2} - 4 ) ^{2}}

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

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

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

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

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

Multipliziere aus

(- 2 x - 5) (x^{2} - 4) - 2 x (2 - 5 x - x^{2}) : 5 x^{2} + 4 x + 20

= \frac{5 x ^{2} + 4 x + 20}{( x ^{2} - 4 ) ^{2}}