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

Lösung

ableitung von

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

\frac{- 3 x ^{2} - 12 x - 3}{( x ^{2} - 1 ) ^{2}}

Schritte zur Lösung

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

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

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

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

Multipliziere aus

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

= \frac{- 3 x ^{2} - 12 x - 3}{( x ^{2} - 1 ) ^{2}}

d er i v a t i v e \frac{( x - 1 )}{( 2 x ^{2} )}

d er i v a t i v e (4 x^{5} + 1)^{3}

\frac{d}{d x} (2 x^{3} - 16 x^{2} - 2 c + 4 d)

d er i v a t i v e g (x) = 3^{2 x + 7}

d er i v a t i v e (x + 5)^{6} \cdot (x^{2} + 2)^{5}