derivative (3x+1)/(x^2+2x+2)

Lösung

ableitung von

\frac{3 x + 1}{x ^{2} + 2 x + 2}

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

Schritte zur Lösung

\frac{d}{d x} (\frac{3 x + 1}{x ^{2} + 2 x + 2})

Wende die Quotientenregel an:

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

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

\frac{d}{d x} (3 x + 1) = 3

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

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

Multipliziere aus

3 (x^{2} + 2 x + 2) - (2 x + 2) (3 x + 1) : - 3 x^{2} - 2 x + 4

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