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

Lösung

ableitung von

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

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

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

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

\frac{d}{d x} ((x + 2) (2 x + 4)) = 4 x + 8

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

= \frac{( 4 x + 8 ) ( - x ^{2} - x - 2 ) - ( - 2 x - 1 ) ( x + 2 ) ( 2 x + 4 )}{( - x ^{2} - x - 2 ) ^{2}}

Multipliziere aus

(4 x + 8) (- x^{2} - x - 2) - (- 2 x - 1) (x + 2) (2 x + 4) : 6 x^{2} + 8 x - 8

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