derivative (x^2+3x-3)/(3x^2-5x+1)

Lösung

ableitung von

\frac{x ^{2} + 3 x - 3}{3 x ^{2} - 5 x + 1}

\frac{- 14 x ^{2} + 20 x - 12}{( 3 x ^{2} - 5 x + 1 ) ^{2}}

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

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

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

\frac{d}{d x} (3 x^{2} - 5 x + 1) = 6 x - 5

= \frac{( 2 x + 3 ) ( 3 x ^{2} - 5 x + 1 ) - ( 6 x - 5 ) ( x ^{2} + 3 x - 3 )}{( 3 x ^{2} - 5 x + 1 ) ^{2}}

Multipliziere aus

(2 x + 3) (3 x^{2} - 5 x + 1) - (6 x - 5) (x^{2} + 3 x - 3) : - 14 x^{2} + 20 x - 12

= \frac{- 14 x ^{2} + 20 x - 12}{( 3 x ^{2} - 5 x + 1 ) ^{2}}