derivative (x^4+6x-1)/(7x^2-6x+1)

Lösung

ableitung von

\frac{x ^{4} + 6 x - 1}{7 x ^{2} - 6 x + 1}

\frac{14 x ^{5} - 18 x ^{4} + 4 x ^{3} - 42 x ^{2} + 14 x}{( 7 x ^{2} - 6 x + 1 ) ^{2}}

Schritte zur Lösung

\frac{d}{d x} (\frac{x ^{4} + 6 x - 1}{7 x ^{2} - 6 x + 1})

Wende die Quotientenregel an:

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

= \frac{\frac{d}{d x} ( x ^{4} + 6 x - 1 ) ( 7 x ^{2} - 6 x + 1 ) - \frac{d}{d x} ( 7 x ^{2} - 6 x + 1 ) ( x ^{4} + 6 x - 1 )}{( 7 x ^{2} - 6 x + 1 ) ^{2}}

\frac{d}{d x} (x^{4} + 6 x - 1) = 4 x^{3} + 6

\frac{d}{d x} (7 x^{2} - 6 x + 1) = 14 x - 6

= \frac{( 4 x ^{3} + 6 ) ( 7 x ^{2} - 6 x + 1 ) - ( 14 x - 6 ) ( x ^{4} + 6 x - 1 )}{( 7 x ^{2} - 6 x + 1 ) ^{2}}

Multipliziere aus

(4 x^{3} + 6) (7 x^{2} - 6 x + 1) - (14 x - 6) (x^{4} + 6 x - 1) : 14 x^{5} - 18 x^{4} + 4 x^{3} - 42 x^{2} + 14 x

= \frac{14 x ^{5} - 18 x ^{4} + 4 x ^{3} - 42 x ^{2} + 14 x}{( 7 x ^{2} - 6 x + 1 ) ^{2}}

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