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

Lösung

ableitung von

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

\frac{10 x ^{2} - 18 x - 3}{( - x ^{2} + 3 x - 3 ) ^{2}}

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

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

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

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

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

Multipliziere aus

(6 x + 1) (- x^{2} + 3 x - 3) - (- 2 x + 3) (3 x^{2} + x) : 10 x^{2} - 18 x - 3

= \frac{10 x ^{2} - 18 x - 3}{( - x ^{2} + 3 x - 3 ) ^{2}}

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