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

Lösung

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

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

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

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

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

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

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

Multipliziere aus

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

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

s l o p e in t erce pt (0, 1), (8, 6)

d er i v a t i v e \frac{3 x - 9}{x}

\frac{d y}{d x} = 7 x

f (x) = ln ∣ x ∣

\frac{d}{d t} (3 t^{2} - 30 t + 72)