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

Lösung

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

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

Schritte zur Lösung

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

Entferne die Konstante:

(a \cdot f)^{'} = a \cdot f^{'}

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

Wende die Quotientenregel an:

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

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

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

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

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

Vereinfache

- 2 \cdot \frac{( - 3 x ^{2} + 3 ) ( x ^{2} + 1 ) ^{3} - 6 x ( x ^{2} + 1 ) ^{2} x ( - x ^{2} + 3 )}{( ( x ^{2} + 1 ) ^{3} ) ^{2}} : - \frac{6 ( x ^{4} - 6 x ^{2} + 1 )}{( x ^{2} + 1 ) ^{4}}

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

x \to \frac{π}{2} lim (cot (x))

n = 5 \sum \infty \frac{1}{2 ^{n}}

\frac{d}{d x} (7 x^{2} ln (x))

f (x) = - arctan (x)

s im pl i f y x 5^{2 x}